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GREENSBOKO,  N.  C: 
PUBLISHEJP  BY  STERLING,  CAMPBELL  &  ALBRIGHT. 

1  s  crs . 


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PERKINS  LIBRARY 

Uuke   University 


Kare  Dooks 


TRINITY  COLLEGE  LIBRARY 

DURHAM,  N.  C. 
1903 


Gift  of  Dr.  and  Mrs.  Drcd  Peacock 


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O  XT  H     O  "W  M 


SCHOOL  ARITHMETIC. 


B  Y 


S.  LA-jSTDER,  a.,  m. 


)0 *•* 

kO  GREENSBORO,  N.  C: 

18  6  3 


PUBLISHED  BY  STERLING,  CAMPBELL  &  ALBRIGHT. 

Richmond,  Va.,  W.  Harorave  White. 


Entered  according  to  Act  of  Congress, 
in  the  year  I860, 
By    S.    lander, 
In  the  Clerk's  Office  of  tlie  District  Court  of  the  Con- 
federate States,  for  the  District  of  Cape  Fear, 
North-Carolina. 


i 

'J 

4i 


P  11  E  F  A  C  E  . 


In  presenting  to  the  public  perhaps  the  first  Arithmetic 
whose  authorship  and  publication  belong  exclusively  to  the 
Confederate  States,  I  call  attention  to  the  following  as  its 
leading  characteristics. 

J .  The  pupil  is  furnished  with  a  model  for  each  class  of 
operations,  by  which  he  may  know  precisely  what  kind  of 
explanation  is  required  of  him  as  he  recites. 

2.  The  distinction  between  abstract  and  concrete  num- 
bers is  carefully  kept  up  throughout  the  whole  book. 

3.  The  Tables  of  Relations  of  Concrete  Numbers  are  un- 
usually full  and  convenient. 

4.  The  problems  are  designed  to  call  into  exercise  the 
pupil's  practical  common  sense,  as  well  as  to  assist  him  in 
acquiring  a  correct  knowledge  of  Arithmetic. 

5.  The  results  of  about  two-thirds  of  the  problems  are 
given  :  those  of  the*  remainder  are  omitted,  and  a  few  of 
those  given  contain  intentional  errors,  to  test  the  pupil's 
self-reliance. 

6.  Progressions,  and  Mensuration,  and  the  ordinary 
methods  of  extracting  roots,  are  excluded  entirely,  because 
they  lie  beyond  the  province  of  Arithmetic. 

7.  No  space  is  wasted  by  the  insertion  of  questions  on 
the  text.  A  teacher  who  can  not  instruct  without  the  help 
of  questions,  will  succeed  but  poorly  even  with  them. 


IV  PREFACE. 


8.  The  discouraging  contradictions  -which  are  so  nume- 
rous in  ''our  best  Arithmetics/'  have  beew  sedulously 
avoided. 

For  the  neat  appearance  of  the  book  I  am  much  indebt- 
ed to  my  publishers,  who  have  spared  neither  pains  nor  ex- 
pense to  bring  it  out  creditably ;  and  I  am  under  especial 
obligations  to  my  friend  Prof.  Theo.  F.  Wolle,  of  Edge- 
worth  Female  Seminary,  without  whose  constant  vigilance 
in  revising  the  sheet.'?;  no  approximation  to  its  present  ac- 
curacy could  have  been  attained. 

I  invite  my  fellow-teachers  to  try  the  book  by  the  unly 
<ure  test,  the  test  of  the  school-room  ;  and  I  will  thank- 
fully receive  any  propositions  of  improvement  which  their 
examinations  may  suggest. 

Our  Own  Primary  Arithmetic  will  follow  this  as  soon  a.< 

possible.  ' 

S.  LANDER. 
LiNCOLNTON,  N.  C,  August  Ij  1863, 


6  0  N  T  E  N  T  S 


Introductiou, v      i 

Arabic  Notation, , -^ 

Roman  Notation , 1  ♦^ 

Operations,  • 

Addition  of  Abstract  Integers, 17 

Subtraction  of  Abstract  Integers, 2r, 

Multiplication  of  Abstract  Integers. 31 

Division  of  Abstract  Integers, 41 

Contraction  in  Addition, 54 

Contraction  in  Subtraction, 5:") 

Contractions  in  Multiplication, 5() 

Contractions  in  Division,. Gf) 

Oeneral  Principles  of  Division, 72  , 

Measures  and  Multiples, 82 

Prime  Factors, ■• ■  •  •  • 99 

Involution, 100 

Evolution, lOi 

Grreatest  Common  Measurv; 103 

Least  Common  Multiple, •-• 10(5 

<'ommon  Fractions, lOD 

Reduction  of  Common  Fractions, 12(' 

Addition  of  Common  Fractions, 131 

Subtraction  of  Common  Fractions, 134 

Multiplication  of  Common  Fraction.^ 139 

Division  of  Common  Fractions — 146 

Cancellation, 15r' 

Decimal  Fractions, — Notation,... 154 

Addition  of  Decimal  Fractions,.... 159 

Subtraction  of  Decimal  Fractions, 161 

Multiplication  of  Decimal  Fractions, 162 

Division  of  Decimal  Fractions, 164 

Contraction  in  Multiplication, 167 

Contraction  in  Division 168 


Tl  CONTENTS. 


Relations  of  Common  and  Decimal  Fractions, §169 

Concrete  Numbers, — Relations, 177 

Addition  of  Concrete  Numbers, 178 

Subtraction  of  Concrete  Num.bers, 180 

Multiplication  of  Concrete  Numbers,,  c 182 

Division  of  Concrete  Numbers, 184 

Reductipn  of  Concrete  Numbers, 186 

Compound  Numbers, , . , 199 

Addition  of  Compound  Numbers, 200 

Subtraction  of  Compound  Numbers, 201 

Multiplication  of  Compound  Numbers, 202 

Division  of  Compound  Numbers, ., 208 

Aliquot  Parts ;   or,  Practice, 205 

Contraction  in  Multiplication, 208 

Contraction  in  Division, 209 

Ratio, 210 

Simple  Proportion, 2 1 G 

Compound  Proportion, 224 

Partitive  Proportion  ;  or.  Fellowship, 229 

Conjoined  Proportion  ;  or.  The  Chain  Rule, 232 

Percentage, i 233 

Jnterest,. ..,......; 242 

Partial  Payments, c 250 

Compound  Interest,. 251 

Discount,. 252 

Bank  Discount,... , 255 

Average, ^61 

Alligation  Medial, 264 

Alligation  Alternate,.*. _ 266 

liquation  of  Payments,.. 268 


ARITHMETIC. 


INTRODUCTION. 


§  1 .  Arithmetic  is  the  science  of  numbers. 

^  2.  A  unit  is  any  single  tiling  f  as,  one,  one  dollar. 

§  8.  A  number  is  a  collection  of  units  ;  as,  three,  two 
<lollars,  four  men. 

§  4.  An  abstract  number  is  one  whose  unit  is  not  speci- 
tied  ;  as,  two,  forty,  seventy-one,  eight. 

§  5.  A  concrete  number  is  one  whose  unit  is  specified : 
us,  ten  dollars,  forty  men,  seventy-one  bales,  eight  books. 

§  6.  Two  or  more  numbers  are  similar  when  they  have 
the  same  unit ;  as,  two,  five,  and  seventy ;  three  men  and 
six  men. 

§  7.  Two  or  more  numbers  are  dissimilar  when  they  have 
different  units  ;  as,  two,  five  dollarst,  seventy  men,  three 
books. 

iVb^e.— All  abstract  numbers  are  similar. 

§  8.  A  compound  number  is  a  concrete  number  expressed 
in  two  or  more  denominations;,  as,  three  dollars,  fifty  cents; 
ten  hogsheads,  forty  gallons,  three  gills ;  ten  miles,  seven 
furlongs,  seventeen  rods. 


§9  ARABIC    NOTATION, 


ARABIC  NOTATION. 


^  9.  Notation  is  the  luethod  of  expressing  numbers  \>y 
figures.  The  Ar'abic  system,  which  is  the  one  in  common 
use,  is  called  also  the  decimal  system,  partly  because  it 
employs  ten  figures.  These  figures  are  :  0  naught  or  zero, 
1  one,  2  two,  3  three,  4  four,  5  five,  6  six,  7  seven,  8  eight, 
9  nine. 

The  figure  0  is  used  to  fill  vacant  places^  and  is  omitted 
in  reading.  » 

§  10.  A  ten  is  a  collueiioAi  often  units,  and  is  called  :i 
unit  of  the  second  order. 

A  hundred  is  ten  tens,»or  one  hundred  units,  and  is  called 
a  unit  of  the  third  order. 

A  thousand  is  ten  hundreds,  or  one  thousand  units ;  and 
is  called  a  unit  of  ihQfourtli  order. 

So,  ten  units  of  any  order  jiiake  one  of  the  next  higher. 

§  11.  A  single  figure  denotes  units;  as,  3,  three  units. 
5,  five  units,  7,  seven  units. 

§  12.  When  two  figures  are  written  together,  the  one  on 
the  right  denotes  units,  and  the  rthertens;  as,  23,  two  tcn.^- 
and  three  units,  that  is,  twenty-three  units ;  34,  three  t^wi^ 
and  four  units,  that  is,  thirty-four  units,  or,  simply,  thirty- 
four. 

Read  27,  66,  73,  37,  84,  48,  99,  ^6,  43,  55,  79,  80,  KS. 

§  13.  When  three  figures  are  written  together,  the  one 
on  the  right  denotes  units,  the  next  tens,  and  the  other 
hundreds  ;  as,  123,  one  hundred,  two  tens,  and  three  units, 
or,  one  hundred  and  tw^enty- three  ;  321,  three  hundred  and 
twenty -one  ;  132,  one  hundred  and  fliirty-two  ;  402,  four 
hundred  and  two. 


ARABIC   NOTATION.  §15 

Read  647,  864,  420,  301,  753,  587.  357,  735,  608,  740, 
047,  306,  700,  609,  069,  009,  290,  391,  001. 

§  14.  When  more  than  three  figures  are  written  together, 
they  arc  separated  into  periods  of  three  figures  each,  begin- 
ning at  the  right ;  find^  in  each..period,  the  three  figures  de- 
note respectively  units,  tens,  and  hundreds,  of  that  period. 

§  15.  The  names  of  the  periods  and  their  order  from 
right  to  left  arc  given  in  the  following 


3       a        o        o 


■X3 

'rJ 

Ph 

H 

a» 

<y 

CO 

tjj 

o 

c 

-3 

r— * 

</, 

• 

cr 

fl 

a 
o 

»— 1 

a 
o 

O 

t! 

C3 

O 

'3 

O 

't, 

rA 

1-^ 

C 

a 

a 

E-i 

ca 

EH 

t^ 

123,004,500,060,000,000,700,089,000,897,060,000,543,210. 

Thfe  above  number  is  read, — one  hundred  and  twenty- 
three  duodecillions,  four  undeeillions,  five  hundred  decil- 
iious,  sixty  nonillions,  seven  hundred  scxti-llions,  eighty- 
nine  quintillions,  eight  hundred  and  ninetj -seven  trillions, 
.-ixty  billions,  five  hundred  and  forty-three  thousand,  two 
hundred  and  ten. 

BuLE  FOii  READING  i»UMBERS.—>S'<rjDamfe  the  jicjiivei^  into 
periods  of  three  figures  each,  beginning  at  the  right;  then, 
heginning  at  the  left,  read  each  period  as  if  it  stood  alone, 
•md  pronounce  the  7i(ime  of  the  period  after  reading  it, 

Kead  the  following  :  12^  21,  37,  86;  793,  842,  209,  319; 
1^346,  7907,  5432,  8642,  4C04,  1861,  1775  ;  24608,  13579, 
10724,  40047,  78009;  475213,  570903,  400101,  300003; 
1230456,2040608,7901035,8000005  ;  70083790,245000542, 
5430102046,  146070080009,  9000800706040,  600005000 r 

0 


§15  ABSTRACT    NUMBERS. 

Write  tbe  following  numbers  in  figures  : 

35.  Seventy-four. 

36.  Four  hundred  and  forty-eight. 

37.  Five  thousand,  three  hundred  and  ninety-seven. 

38.  Sixty  thousand,  and  seventeen.    ^ 

39..  Seven  hundred  and  forty  thousand,  eight   hundred 
and  forty-one. 

40.  Eight  millions,  sev-enty  thousand,  and  seventy-nine. 

41.  Ninety-four  millions,  sixteen  thousand,  four  hundred 
and  fourteen. 

42.  Thre«  hundred  millions,  and  three. 

43.  Two  billions,  two  millions,  two  thousand,  and  two. 

44.  Ten  billions,  ten  millions,  ten  thousand,  and  ten. 

45.  Nine  hundred  and  twenty-five  billions,  eight   thou- 
sand, and  sixteen. 

46.  Eight  trillions,  seven   billions,    sixty  millions,   five 
thousand,  and  four. 

47.  Seventy  trillions,  eighty -nine  millions,  and  twenty- 
one. 

48.  Six  hundred  and  forty-two  trillions,    three  hundred 
thousand. 

49.  Fifty-three  quadrillions,  eleven  billions,  and  seventy- 
three. 

50.  Four  hundred  and  four    quintillions,   two   hundred 
and  two  millions. 

51.  Thirty  sextillions,  forty  quintillions,  fifty   trillions, 
six  hundred  and  two. 

52.  Two  octillions,  four  quadrillions,  six  hundred    and 
eight  thousand. 

53.  Ten  decillions,  twelve  nonillions,  fourteen  millions, 

and  ninety-nine. 

54.  Nine  ijonillions,  ten*  millions,  and  twenty-seven. 

10 


ROMAN    NOTATION.  §16 


ROMAN  NOTATION. 


yj)  IG.  The  Koman  Notation  employs  the  following  seven 
letters  :  I  one,  V  five,  X  ten,  L  fifty,  C  one  hundred,  D  five 
hundred,  and  M  one  thousand. 

All  integral  numbers  may  be  denoted  by  combining  these 
letters  accordiug  to  the  following  rules  : 

1.  Any  letter  doubled  denotes  twice  its  simple  value  ; 
tripled  denotes  three  times,  and  so  on.  Thus,  11=2, 
XX=20,  CCC  =  300. 


'J 


2.  If  a  letter  of  less  value  is  placed  after  one  of  greater 
value,  tlie  less  is  to  be  added  to  the  greater.  Thus,  ¥1=6, 
XV=15,  CGL=250. 

3.  If  a  letter  of  less  value  is  placed  he  fore  one  of  greater 
value,  the  less  is  to  be  subtracted  from  the  greater.  Thus, 
iy==4,  XC=90,  CD  =  400. 

4.  If  a  letter  of  less  value  is  placed  hciwetni^o  of  greater 
value,  the  less  is  to  be  subtracted  from  the  sum  of  the  other 
two.     Thus,  XIX=  19,  Xiy  =  U,  XCIX=99. 

5.  A  dash  placed  over  a  letter  multiplies  its  value  by 
1000.     Thus,  "1=50000/0  =  100000. 

The  above  rules  are  sufficiently  exemplified  in  the  fal- 
lowing 


T.4lI5LiJEC. 

I--=l 

XI-=11 

XXI=21 

C=100 

II----2 

Xll-^]  ■-. 

XXII=22 

CC=200 

111=3 

xiri=i3 

:vXIII=2.3 

CD=400 

IV=^1 

XlVc^li 

XXX=30 

D=500 

V=5 

XV=15 

XL=;:40 

DC=600 

VI=6 

XVI^IG 

L=50 

M=1000  ^ 

VIIr=7 

XVII=17 

LX=60 

MC=1100  ' 

vni=8 

XVIII=18 

LXX=70 

MM=2000 

IX=9 

XIX=19 

LXXX=80 

11=1000000 

X=^10 

XX=::20 

XC=90 

U 

MDCCCLXIII:^1808 

§1' 


ABSTRACT    NUMBERS. 


OPEKATIONS. 


There  are  four  operations  in  Arithmetic  ;  Addition,  Sub- 
traction, Multiplication,  and  Division.  We  will  explain 
these  operations  in  succession,  first  with  reference  to  ab- 
stract numbers,  and  afterwards  with  reference  to  concrete 
numbers. 


ADDITION  OF  ABSTRACT  NUMBERS. 


§  17.  Addition  is  the  operation  of  finding  one  number 
equal  to  several  other  numbers  put  together. 

§  18.  The  result  of  addition  is  called  the  sum  of  the  num- 
bers added.     Thus,  10  is  the  sum  of  6  and  4. 

§  19.  The  sign  of  addition^  -,'-,  is  read  jo/t^s.  When  pla- 
ced before  a  number,  it  denotes  that  it  is  to  be  added  to  any 
other  additive  number  with  which  it  is  connected.  Thus, 
()-f4,  6  plus  4,  denotes  four  added  to  six. 

§  20.  l^hQ  sign  of  equality^  =,  is  read  i^  equal  to.  When, 
placed  between  two  expressions  it  denotes  that  they  are  equal 
to  each  other.     Thus,  6+4=10     Also,  7+4  +  3  =  8  +  6. 


1  aud  0  are  2 

3  and  0  are  3 

4  and  0  are  4 

5  aud  0  are  ;' 

'1  and  1  are  3 

3  and  1  are  4 

4  and  1  are  5 

5  and  1  are  G 

li  and  2  are  4 

3  and  2  are  5 

4  and  2  are  6 

5  and  2  are  7 

'J  and  3  are  5 

3  and  3  are  6 

4  and  3  are  7 

5  and  3  are  8 

2  and  4- are  0 

3  and  4  are  7 

4  and  4  are  8 

5  and  4  are  9 

li  and  5  are  7 

3  and  o  f.re  8 

4  and  5  are  9 

5  and  5  are  10 

il  and  6  are  8 

3  and  C  are  9 

J  4  and  6  are  10 

5  and  6  are  11 

2  and  7  are  9 

3  and  7  are  10 

1  4  and  7  are  11 

5  and  7  are  12 

2  and  8  are  10 

3  and  8  are  11 

4  and  8  are  12 

5  and  8  are  IS 

2  and  0  are  11 

3  and  0  are  12 

4  and  9  are  13 

5  and  9  are  14 

12 


ADDITION    OF   INTEGERS 


§22 


G  and  0  arc  G 
G  and  1  nre  7 
6  and  2  nre  8 
G  and  3  are  9 
G  and  4  are  10 
G  and  5  are  11 
G  and  G  are  12 
G  and  7  are  13 
G  and  8  are  14 
<i  and  9  are  15 


7  and 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and 
7  and 


0  are  7 

1  are  8 

2  are  9 

3  arc  10 

4  are  11 

5  are  12 
G  are  13 

7  are  14 

8  are  15 

9  arc  16 


8  and 
8  and 
8  and 
8  and 
8  and 
8  and 
8  and 
8  and 
8  and 
Sand 


0  are  ^ 
T  are  9 

2  are  10 

3  arc  1 1 

4  are  12 

5  are  13 

6  are  14 

7  are  15 

8  are  IG 

9  are  17 


9  andO 
9  a4id  1 
9  and  2 
9  and  3 
9  and  4 
9  and  5 
9  and  6 
9  and  7 
9  and  8 
9  and  9 


are  U 
are  10 
are  1 1 
are  12 
are  13 
are  14 
are  15 
are  IG 
are  17 
are  18 


Note. — Let  the  above  table  bo  thoroughly  memorized  before  the 
pupil  advances  farther. 

Ex.  1.  Add  togetkcr  102741,  42102,  and  3050. 

102741  ^  21.  Model.— 2  and  1  are  3;  5  and  4  arc 

42102         9  ^   I  ^j^(2  7  arc  8  ;  3  and  2  are  5,  and  2  are 
3050         7    4     i_     The  sum  is  147893. 


147893 


Note. — Let  the  teacher  see  to  it  that  the  pupil  recites  precisely 
according  to  the  model  ttoth  here  and  -wherever  a  model  is  given. 

Explanation. — First,  the  numbers  are  arranged  with 
units  of  the  same  order  in  the  same  column.  Then,  begin- 
ning at  the  right,  the  numbers  in  each  column  are  added  to- 
gether, and  the  sum  is  placed  underneath  in  the  same  column . 

2.  Addtogetlier23456, 10203,  and  56030.     Sum,  89689. 

3.  Find  the  sum  of  120242,  334124,  and  224612. 

Note. — Let  the  pupil  first  say,  -'Add  the  numbers  together/" 
and  then  proceed  as  in  the  model. 

4.  What  is  the  sum  of  2400, 1505,  and  3074  ?    Ans.  6979. 

5.  Add  270,  102,  314,  and  301  together.  Sum,  987. 

6.  Add  together  94085,  16275,  and  3367. 

94085  §  22.  Model.— 7  and  5  are  12,  and  5 

1627S         are  17,  set  down  7  ;  1  and  6  are  7,  and  7 

3367         are  14,  and  8  are  22,  set  down  2 ;  2  and  3 

Sum,  113727         are  5,  and  2  are  7  ;  3  and  6  are  9,  and  4 

are  13,  set  down  3 ;  1  and  1  are  2,  and  9 

are  11,  set  down  11.     The  sum  is  113727. 

13 


§22  ABSTRACT    NITMBERS. 


,  Explanation. — After  arranging  tlie  numbers  as  in  §  21, 
the  sum  of  tlie  column  of  units  is  found  to  be  17  units,  that 
is,  1  ten  and  7  units ;  hence,  the  7  is  placed  under  the  col- 
umn of  units,  and  the  1  is  afterwards  added  in  with  the 
column  of  tens.  The  sum  of  the  column  of  tens  are  22  tens, 
that  is,  2  hundreds  and  2  tens ;  hence,  the  right  hand  2  is 
placed  under  the  column  of  tens,  and  the  other  2  is  added 
in  with  the  column  of  hundreds.  The  sum  of  the  column 
of  hundreds  is  7  hundreds,  and  the  7  is  placed  underneath 
in  that  column.  The  sum  of  the  column  of  thousands  is 
13  thousands,  that  is,  1  ten-thousand  and  3  thousands ; 
hence,  the  3  is  placed  in  the  column  of  thousands,  and  the 
1  is  added  in  with  the  column  of  ten-thousands.  The  sum 
of  the  column  of  ten*  thousands  is  11  ten-thousands,  that  is, 
1  hundred-thousand  and  1  ten-thousand ;  hence,  the  right 
hand  1  is  placed  in  the  column  of  ten-thousands,  and  the 
other  1  in  the  place  of  hundred-thousands. 

Rule. — Arrange  the  numbers  icith  units  of  the  s<^nie.  or- 
der in  the  same  column. 

Beginning  at  the  right,  find  the  sum  <?/  each  column  ;  if 
this  sum  is  expressed  hy  one  figure,  set  it  down  under  the  col- 
umn; but  if  it  is  expressed  by  more  than  one  figure,  set  th-e 
right  hand  figure  under  this  column,  and  add  the  remaining 
figure  or  figures  in  with  the  next  column. 

Set  down  the  whole  sum  of  the  last  column. 

Pnoor. — 1,  Add  as  before,  but  begin  at  the  top  of  each 
column. 

Or,  2.  Find  the  sum  of  all  the  numbers  but  one,  and  to 
this  sum  add  the  number  excepted. 

Ex.  7.  Add  together  234,  15G,  987,  and  358.    Sum,  1735. 
8.  Add  together  1020,  304,  66,  and  9.  Sum,  1389, 

14 


ADDITION'    OF   INTEGERS.  §22 


9.  Add  together  2739,  9647,  271,  17,  and  2U50. 

10.  Add  together  169078,  270189,  and  928608. 

Sum,  1367875. 

11.  Add  together  27090,  2709,  2^905,  27,  2709050,  and 
2r0.  Sum,  3010051. 

12.  Find  tha  sum  of  369764,  275863, 10794,  273, 102469, 
and  1861. 

13.  Find  the  sum  of  173594,  240680,  10305,  678,  and 
976531.  Sum,  1401788. 

14.  Find  the  sum  of  97347S25,  89734782,  28973478, 
828973478,  and  98289734.  Sum,  1143319297. 

15.  Find  the  sum  of  1928374560,  192837456,  1928, 
19283745,  1928374,  192837,  and  19283. 

16.  907050301  4-80604020-1-  123123123-=what  ? 

Ans.  1110777444. 

17.  146-f  1375-M3795-f  246820-f  24682=what  ? 

Ans.  2S681b. 

18.  2620-h6202-f7593-4-3694-f  1735=what? 

19.  "What  is  the  sum  of  3426,  9120634,  52714,  9987, 
1137,  and  97579? 

20.  What  is  the  sum  of  26322,  50555,  37684,  898955, 
and  9024  ?  Ans.  1022540. 

21.  WhatisiAie  sum  of  41084,  293347,  9139919,  and 
46552?  Ans.  9520902. 

22.  What  is  the  sum  of  245301,  586642,  51407,  1752, 
71283,  and  42061  ? 

23.  What  is  the  sum  of  10,  105,  1057,  10572,  105723, 
1057234, 10572349,  105723496,  and  1057234968? 

Ans.  1174705532. 

24.  What  is  the  sum  of  135792468,246813579,159483726, 
372684951,  123456789,  896745321,  896453217,  arid 
400500746  ?  Ans.  3231930797. 

15 


§23 


ABSTaACT    NUMBERS. 


SUBTRACTION  OP  ABSTRACT  NUMBERS. 


§  23.  Subtraction  is  the  operation  of  finding  the  differ- 
ence between  two  iiurabers,  by  taking  the  less  from  the 
greater. 

§  24.  The  number  to  he  aubtracted  is  called  the  subtra- 
hend. 

§  25.  The  number  to  he  diminislied  is  called  the  imnn- 
tmd. 

§  2%.  The  residt  of  subtraction  is  called  the  remainder 
or  the  difference. 

§  27.  The  sign  of  subtraction,  — ,  is  read  minus.    When 
placed  before  a  number,  it  denotes  that  it  is  to  be  subtracted 
from  the  number  with  which  it  is  connected.     Thus, 
(>— 4,  6  minus  4,  denotes  4  taken  from  6.     Also,  7—3=4. 

§  28.  The  remainder  is  not  changed  by  increasing   the 
minuend  and  the  subtrahend  equally.     Thus, 
Min.     27      27  +  15=42      274-240=267      27  +  306;=333 
Sub.     16      16+15=31      16+240=256      16  +  306=322 

Kern.  11  n  nn  ~Y\ 


from  1 

leaves  0 

2  from  2 

leaves  0 

from  2 

leaves  1 

2  from  S 

leaves  1 

from  3 

leaves  2 

2  from  4 

leaves  2 

from  4 

leaves  3 

2  from  5 

leaves  3 

from  § 

leaves  4 

2  from  6 

leaves  4 

from  6 

leaves  5 

2  from  7 

leaves  5 

from  7 

leaves  6 

2  from  8 

leaves  G 

from  8 

leaves  7 

2  from  9 

leaves  7 

from  9 

leaves  8 

2  from  10  leaves  8 

from  10  leaves  9 

2  from  11  leaves  9 

from 
from 
from 
from 
3  from 
from 
from 
from 
from 
from 


leaves  0 
leaves  1 
leaves  2 
leaves  3 

7  leaves  4 

8  leaves  5 

9  leaves  6 

10  leaves  7 

11  leaves  8 

12  leaves  9 


16 


SUBTRACTION    OF   INTEGERS. 


§29 


4  from 

4    lenves  0 

i     5  from  6    leaves  0 

i    ^ 

from 

6    leaves  0 

4  from 

5     leaves  1 

j     5  from  6    leaves  1 

i     6 

from 

7    leavea  1 

I  from 

0    leaves  2 

!     5  from  7    leaves  2 

1   ^ 

from 

8    leaves  2 

i  from 

7    leaves  3 

5  from  8    leaves  3 

i     6 

from 

9    leaves  8 

4  from 

8    leaves  4 

5  from  0    leaves  4 

'     6 

from 

10  leaves  4 

It  from 

0    leaves  5 

5  froru  10  leaves  5 

I     6 

from 

11  leaves  6 

1   from 

10  leaves  G 

5  from  11  leaves  6 

;     6 

from 

12  leavei  G 

4  from 

11  leaves  7 

i»  from  12  leaves  7 

:    6 

from 

13  leaves  7 

I  from 

12  leaves  8 

0  from  13  leaves  8 

:     G 

from 

14  leaves  8 

'{  from 

18  leaves  0 

5  from  14  leaves  9 

G 

from 

15  leaves  9 

7   from 

7     leavea  0 

8  from  8    leaves  0     ' 

9 

from 

9    leaves  0 

7  from 

8    leaves  1 

8  from  9    leaves  1     \ 

9 

from 

10  leaves  1 

7  from 

1>    leaves  2 

8  from  10  leaves  2     j 

9 

from 

1 1  leaves  2 

7  from 

10  leave."  3 

8  from  11  leavob  3     ' 

9 

from 

12  leaves  3 

7  from 

1 1  leaves  4 

8  from  12  leaves  4 

9 

from 

13  leaves  4 

7  from 

12  leaves  o 

8  from  13  leave?  5     i 

9 

from 

14  leaves  5 

7  from 

13  leaves  G 

8  from  14  leaves  G     ; 

9 

from 

15  leaves  G 

7  from 

14  leaves  7 

8  from  15  leaves  7     1 

9 

from 

IG  leaves  7 

7  from 

15  loaves  8 

8  from  16  leaves  8     J 

9 

from 

17  leaves  8 

7   from 

IG  leaves  9 

8  from  17  leaves  9     ! 

9 

from 

18  leaves  0 

Mill. 

8ub. 

Rem. 


I.  From  976348  subtract  35127. 

§  29.  Model.— 7  from  8  leaves  1 ;  2 


976348 
35127 

941221 


from  4  leaves  2  ;  1  from  3  leaves  2  ;  5 
from  0  leaves  1  ;  3  from  7  leaves  4 ;  0 
from  9  leaves  9.  The  remainder  is 
941221. 

Explanation— The  subtrahend  is  placed  under  the  min- 
uend, with  units  of  the  same  order  in  the  same  column. 
Then,  beginning  at  the  right,  each  figure  of  the  subtrahend 
is  taken  from  the  corresponding  figmre  of  the  minuend,  and 
the  remainder  is  set  underneath  in  the  same  column. 

2.  From  127936  subtract  14312.  Rem.  113624. 


3.  From  96898  subtract  13456. 

4.  Subtract  864231  from  9557654. 

5.  Subtract  1024370  from  12357799. 


B 


17 


Rem.  123423. 
Rem.  ] 1333429. 


§30  ABSTRACT    NUMBERS. 


6.  Subtract  327739  from  573G47. 

Min.     573647  §30.  Model.— 9  from  17  leaves  8  ;  4 

Sub.      327739         from  4  leaves  0 ;  7  from  16  leaves  9  ;  8 

Rem.    245908         from  13  leaves  5;  3  from  7  loaves  4:  3 

from  5  leaves  2.  The  remainder  is  245908. 

Explanation. — After  arranging  the  numbers  as  in  §  29, 
it  is  required  to  take  9  units  from  7  units  :  this  can  not  be 
done ;  hence,  1  ten,  that  is,  10  units,  is  added  to  the  minu- 
end, giving  17  units,  from  which  9  units  taken  leaves  8  units. 
Then,  because  the  minuend  is  increased  10  units  or  1  ten, 
the  subtrahend  must  be  increased  the  same  amount  (§  28). 
This  gives  4  tens  to  be  taken  from  the  4  tons  of  the  minu- 
end, leaving  0  tens.  Again,  7  hundreds  can  not  be  taken 
from  6  hundreds ;  hence,  1  thousand,  that  is,  10  hundreds, 
is  added  to  the  minuend,  giving  16  hundred:?,  from  which 
7'  hundreds  taken  leaves  9  hundreds.  Then,  because  the 
minuend  is  increased  10  hundreds  or  1  thousand,  the  sub- 
trahend must  be  increased  the  same  amount. 

The  same  kind  of  reasoning  will  explain  the  rest  of  the 
operation. 

Rule, — Place  the  siihtraJiend  under  (lie  niinucnd,  wilh 
units  of  the  same  order  in  the  same  column. 

Beginning  at  the  rights  take  each  figure  of  the  subtrahend 
from  the  corresponding  figure  of  the  minuend. 

If  any  figure  of  the  minuend  is  less  than  the  corresponding 
figure  of  thc^  subtrahend,  add  10  to  this  minuend  figure  ^  and 
add  1  to  the  subtrahend  figure  in  tlie  next  column. 

Proof. — 1.  Add  the  remainder  to  the  subtrahend  ;  the 
sum  will  be  equal  to  the  minuend. 

Or,  2.  Subtract  the  remainder  from  the   minuend ;  the 

difference  will  be  equal  t©  the  subtrahend. 

'   18 


SUBTEACTION  OF  INTEGERS. 


'M 


Ex.  7.  From  S9G  take  307. 

8.  From  1842  take  961. 

9  From  2719  take  1827. 

10.  From  12791  take  9872. 

11.  From  24598  take  20689. 

12.  From  978637  take  97863. 

13.  From  1654278  take  755429. 

14.  Take  678902  from  896454. 
1^.  Take  1724937  from  1963869, 
16'.  Take  23468579  from  60050040. 

17.  Take  9879789  from  9900000. 

18.  Take  7890845  from  lOOOOOOO. 

19.  Miiiuend  =  1234567,  Subtrahend 


Eem.  589. 
Eem.  881. 

Eem.  2919. 
Rem.  3909. 

Rem.  898849. 
Rem.  217552. 

Rem.  36581461. 
Rem.  20211. 


765432. 
Rem.  469135. 

Begin  by  saying,  "Subtract  tlie  Subtrahend  from  the 


Rem.  181765. 

Rem.  7777782. 
Ans.  2887879. 

one  million,  and 
Ans.  999901. 


Note 
Minuend.""  ^ 

20.  Min.=290178,  Sub.  =  108405. 

21.  Sub.=20499,  Min.-~1900623. 

22.  Sub.  =987631,  Min. =8765413. 

23.  12646723-975894^=what? 

24.  2468000— 970053  =  what? 

25.  What  is  the  difference   between 
ninety -nine  1 

Xoic. —  Begin,   "Subtract  the  less  number  from  the  greater.'" 

26.  What  is  the  difference  between  thirty-seven  billions, 
and  eleven  ?  Ans.  36999999989. 

27.  What  is  the  difference  between  nine   thousand   six 
hundred  and  thirteen,  and  five  hundred  and  forty-two  ? 

28.  What  is  the  difference  between  eight  thousand  and 
tw«nty-six,  and  eight  hundred  and  twenty-six?  Ans.  7200. 

29.  What  is  the  difference   between  five  thousand  four 
hundred  and  ninety,  and  seven  hundred  and  sixty-two  1 

19 


|3l  ABSTRACT    NCMBERf. 


^MULTlPLICATrON  OF  ABSTKAOT  NUMBER.S. 

§  ol.  Multiplication  is  the  operation  of  finding  a  num- 
Vier  ■which  shall  contain  one  of  two  given  num})ers  as  many 
times  as  there  are  units  in  the  other. 

Thus,  o  times  6  are  18  :  here  6  is  multiplied  by  3,  be- 
'■auae  18  contains  6,  3  times. 

§  32.  The  number  io  he  multiplied  is  called  the  muUi' 
f)Hcand. 

§  33.  The  inultiplying  number  is  called  the  juuftiplier, 

§  84.  The  result  of  multiplication  is  called  the  product. 

§  35.  Either  the  multiplicand  or  the  multiplier  is  called 
a  factor  of  the  product,  and  thej  both  are  called  itfi  factors. 

In  general,  one  number  is  a  factor  of  any  other  number 
which  contains  it  on  exact  number  of  timea. 

Thus,  3  is  a  factor  of  18  ;  4  is  a  factor  of  12,  or  of  20  ; 
5  is  a  factor  of  10,  of  15,  of  30,  or  of  45. 

§  36.  The  si'^n  of  multiplication^  X,  when  placed  between 
two  numbers,  denotes  that  one  of  them  is  to  b«  multiplied 
by  the  other.  It  is  read  times,  when  placed  after  the  mul- 
tiplier, and  multiplied  by,  when  placed  after  th«  multipli- 
cand. Thus,  to  denote  that  6  ia  to  be  multiplied  by  3,  we 
may  Miy,  3x6,  3  times  6,  or  6x3,  6  multiplied  by  3.  To 
denote  the  successive  multiplication  of  more  than  two  num- 
bers, periods  are  used.  Thus,  2.3.5  =  30.  2  times  3  times 
f)=«30. 

§  37.  The  product  of  any  two  abstract  factors  ia  the 
i^ame,  no,  matter  which  is  used  as  multiplier.  Thus,  3  x6s— 
«X3==18;  4x5-=5x4=20;  10x8-=8xl0=«80. 

20 


MULTIPLICATION    OF  INTEGERS. 


§o7 


IVIUI-iTlJeLlCATIOlV     T^VBLE. 


Oace 

0 

is 

0 

Twice 

0 

are 

0 

3 

times 

0  are 

It 

Once 

\ 

is 

1 

Twice 

1 

are 

i> 

3 

times 

1  are 

o 

Once 

2 

is 

2 

Twice 

o 

Ml 

are 

4 

3 

times 

2  are 

6 

Once 

3 

is 

3 

Twice 

3 

are 

G 

3 

times 

3  are 

9 

Once 

4 

is 

4 

Twice 

4 

are 

8 

3 

times 

4  are 

12 

'  hice 

T) 

is 

5 

Twice 

5 

arc 

10 

3. 

times 

5  are 

15 

Once 

« 

is 

G 

Twice 

G 

are 

12 

'"> 

times 

G  are 

18 

Once 

4 

is 

7 

Twice 

7 

are 

14 

8 

times 

7  are 

21 

Once 

8 

IS 

8 

Twice 

8 

are 

IG 

o 

times 

8  are 

24 

Ouce 

9 

is 

9 

'     Twice 

9 

are 

18 

time3 

0  are 

27 

Once 

10 

is  10 

Twice 

10 

are 

20 

8 

times 

10  are 

30 

Once 

11 

is  11 

Twice 

11 

are 

22 

times 

11  are 

33 

Once 

12 

t 

is  1 

o 

Twice 

12 

are 

24 

1     3 

times 

12  are 

36 

4 

times 

0 

are 

0 

5  times 

0 

are 

0 

6 

times 

0  are 

0 

4 

times 

1 

are 

4 

5  times 

1 

are 

5 

6 

times 

"1  are 

6 

4 

times 

•) 

are 

8 

5  times 

O 

are 

10 

6 

times 

2  are 

12 

4 

timei) 

3 

arQ 

12 

0  times 

3 

are 

15 

6 

times 

3  are 

18 

4 

times 

4 

are 

IG 

5  times 

4 

are 

20 

6 

times 

4  are 

24 

4 

times 

5 

are 

20 

5  times 

5 

are 

25 

G 

times 

5  are 

3ft 

4 

timet 

G 

jire 

24 

5  times 

6 

are 

30 

0 

times 

G  are 

36 

4 

timts 

7 

are 

28 

5  times 

7 

are 

35 

6 

times 

7  are 

42 

4 

times 

8 

are 

32 

5  times 

8 

are 

40 

G 

times 

8  arc 

48 

4 

times 

■  9 

are 

36 

5  times 

0 

are 

45 

6 

times 

9  are 

64 

4 

times 

10 

are 

40 

5  times 

10 

are 

50 

6 

times 

10  are 

60 

4 

times 

11 

are 

44 

5  times 

11 

are 

55 

6 

times 

11  are 

GG 

4 

times 

12 

are 

48 

5  times 

12 

are 

60 

6 

times 

12  are 

72 

7 

time:.> 

0 

arc 

0 

8  times 

0 

arc 

0 

0  i 

iraes 

0  are 

0 

7 

times 

1 

are 

7 

8  times 

1 

arc 

8 

9  t 

imes 

1  are 

9 

y 

times 

2 

are 

14 

8  times 

are 

16 

9  t 

imes 

2  are 

18 

^ 

times 

8 

arc 

21 

8  times 

8 

are 

24 

9  t 

imes 

3  are 

27 

T 

times 

4 

are 

28 

8  times 

4 

are 

82 

9  t 

imes 

4  ar« 

36 

7 

times 

T) 

are 

35 

8  times 

5 

are 

40 

9  t 

imes 

5  are 

45 

7 

tim(;s 

G 

aio 

42 

3  times 

G 

are 

48 

9  t 

imes 

6  are 

54 

7 

times 

7 

are 

40 

8  times 

7 

are 

53    1 

9  t 

iini>8 

7  are 

{Vd 

*T 

times 

8 

jire 

oG 

8  times 

8 

are 

64 

9  t 

imes 

8  are 

7a- 

"7 

times 

0 

are 

68 

8  times 

9 

are 

72 

9  t 

imes 

9  arc 

81 

»T 

times 

10 

are 

70 

8  times 

le 

arc 

80 

9  t 

imes 

10  are 

90 

T 

times 

11 

are 

77 

8  times 

11 

arc 

88 

9  t 

imes 

11   are 

9& 

7 

tiraos 

12 

arc 

»4 

8  times 

12 

arc 

9G 

9  t 

imes 

[2  are 

108 

21 


ABSTRACT    NUMBERS. 


10 
10 


10  times 
10  times 
10  times 
10  times 
times 
times 
10  limes 
10  times 
10  times 
10  times 
10  times 
10  times 
10  times 


0  are 

1  are 

2  are 

3  a,re 

4  are 

5  are 
G  are 

7  are 

8  are 

9  are 

10  are 

11  are 

12  are 


0 

11  til 

10 

n  ti 

20, 

11  til 

SO 

11  ti 

40 

11  ti 

60 

11  ti 

GO 

11  ti 

70 

11  ti 

8§ 

11  ti 

90 

11  ti 

100 

11  ti 

110 

11  ti 

120 

11  ti 

mes 

0  are 

0 

12 

times 

0 

are 

0 

mes 

1  are 

11 

12 

times 

1 

are 

12 

mes 

2  are 

22 

times 

2 

are 

24 

mes 

3  are 

33 

12 

times 

3 

are 

36 

mes 

4  are 

44 

12 

times 

4 

are 

48 

mes 

5  are 

55 

12 

times 

6 

are 

60 

imes 

6  are 

66 

12 

times 

6 

are 

72 

mes 

7  are 

77 

12 

times 

7 

are 

■84 

imes 

8  are 

88 

12 

times 

8 

are 

96 

mes 

9  are 

99 

12 

times 

9 

are 

108 

mes 

10  are 

110 

12 

times 

10 

are 

120 

mes 

11  are 

121 

1  ^ 

X.JU 

times 

11 

are 

132 

imes 

12  are 

132 

12 

times 

12 

are 

144 

Ex.  1.  Multiply  24307  by  3. 
Multiplicand,  24307  §  33.  Model.— 3   times   7    are 

Multiplier,       3         21,  set  down  1  ;  3  times  0  are  0, 

Product,  72921         and  2  are  2;  3  times  3  are  9 ;  3 

times  4  are  12,  set  down  2;  3 
tfimes  2  are  6,  and  1  are  7.     The  product  is  72921. 

Explanation. — Tke  smaller  factor  is  placed  under  the 
larger.  Then,  beginning  at  the  right,  each  figure  of  the 
upper  number  is  taken  3  times,  the  right  hand  figure  of  each 
product  is  set  down,  and  the  remaining  figure,  if  any,,  is 
added  to  the  next  product.  3  times  7  units  are  21  units, 
that  is,  2  tens  and  1  unit ;  hence,  1  unit  is  set  in  the  units' 
]jlace,  and  2  tens  are  added  to  the  product  of  the  tens. 

2.  Multiply  24307  by  40. 
Multiplicand,  24307  §  39^  Model.— 4   times'  7   are 

Multiplier,       40       28,  sef^down  8  ;  4  times  0  are    0, 

Product,  972280       aad  2  are  2  ;  4  times  3  are  12,  set 

down  2 ;  4  times  4  are  16,  and  1 
are  17,  set  down  7  ;  4  times  2  are  8,  and  1  are  9 :  annex  0. 
The  produ<;t  is  972280. 

Explanation. — Since  10  units  of  any  order  make  one  of 
t4e  next  order  on  the  left,  any  number  is  rflultiplied  by  10 

22 


MULTIPLICATION    OF    INTEGERS. 


§40 


by  merely  moving  eacli  of  its  figures  one  place  to  the  left, 
and  putting  a  0  in  the  place  of  units.  Hence,  to  multiply 
by  40,  each  figure  of  the  product  by  4  is  set  one  place  to 
the  left,  and  the  units'  place  is  filled  with  a  0. 

3.  Multiply  24307  by  43. 


Multiplicand, 
Multiplier, 

1st  partial  prod. 
2nd'  partial  prod. 

Product, 


24307 
43 

72921 

97228 

1045201 


§  40.  Model. — 3  times 
7  are  21,  set  down  1 ;  3 
times  0  are  0,  and  2  arc  2 ; 

3  times  3  are  9  ;  3  times 

4  are  12,  set  down    2  ;  3 
times  2  are  6,  and    1    are 

7  :— 4  times  7  are  28,  set  down  8  under  2 ;  4  times  0  are  0, 
and  2  are  2  ;  4  times  3  are  12,  set  down  2 :  4  times  4  are 
IG,  and  1  are  17,  set  down  7  ;  4  times  2  are  8,  and  1  are  9. 
Add  the  partial  products  :  1 ;  8  acd  2  are  10,  set  down  0  ; 

1  and  2  are  3,  and  9  are  12,  set  down  2  ;  1  and  2  are  3,  and 

2  are  5;  7  and  7  are  14,  set  down  4  ;  1  and  9  arc  10,  set 
down  10.     The  product  is  1045201. 

Explanation. — The  upper  nuniLur  is  multiplied,  first 
by  3,  as  in  §  38,  and  then  by  40,  as  in  §  39,  except  that  the 
0  at  the  right  is  omitted,  as  being  unnecessary,  since  the 
several  figures  can  be  placed  in  their  proper  columns  with- 
out it.  It  must  be  remembered,  however,  that  the  second 
partial  product  is  not  97,228,  but  972,280. 

4.  Multiply  3047  by  246279. 

5.  Multiply  794378  by  4608. 


Multiplier, 
Multiplicand, 


Product, 


246279 ; 
3047 

T723953 
985116 

738837 


750412113 


Multiplicand, 
Multiplier, 


!  Product, 


794378 
4608 

6355024 
4766268 
3177512 

3660493824 


23 


§40 


ABSTRACT    NUMBERS, 


Rule. — 1.  When  either  factor  contains  hut  one  valuable 
figure.  Set  the  smaller  factor  under  the  lafger.  Beginning 
at  the  right,  multiply  each  figure  of  the  upper  number  by  the 
lower  number,  set  doicn  the  right  hand  figure  of  the  product, 
and  add  the  remaining  figure,  if  any,  to  the  next  product ; 
but  set  doivn  the  whole  of  (he  last  product. 

2.  When  the  smaller  factor  contains  more  than  one  valua- 
ble figure.  Set  it  under  the  larger  ;  multiply  the  upper  fac- 
tor by  each  figure  of  the  lower,  setting  the  firat  figure  of  each 
partial  product  under  tin-  .'n-i'iijli^ing  figure  which  produced 
it,  and  add  the  partial  products  together  in  thai  order. 

Proof. — Multiply  the  lower  factor  by  tlie  upper. 


Ex.  6.  Multiply  3469  by  3. 

7.  Multiply  4'by  268.  ' 

8.  Multiply  45274  by  5. 

9.  Multiply  56295  by  6. 

10.  Multiply  75397  by  7. 

11.  Multiply  9  by  98765. 

12.  Multiply  21179  by  27. 

13.  Multiply  97825  by  34. 

14.  Multiply  86906  by  45. 

15.  Multiply  279862  by  52. 

16.  Multiply  192837  by  67. 

17.  Multiply  293705  by  75. 

18.  Multiply  246835  by  83. 

19.  1964326  x98=«  what?  Ans.  192503948. 

NoU. — Begin,   "  Maltiply  the  first  number  by  the  second.'" 

20.  What  is  the  product  of  2975x375  ?     Aus.  1 1 15625. 

21.  What  is  the  product  of  3047x287  ? 

24 


Prod.  10407 
Prod.  1072. 

Prod.  226370. 

Prod.  527779. 
Prod.  §88885. 

Prod.  3326050. 
Prod.  3910770 

Prod.  12920079. 
Prod.  22027875. 


DIVISION    OF    INTEGERS.  §48 

22.  What  isthe  product  of  40535x403?  Ans.  19983755. 

23.  What  isthe  product  of  4-027x4027?  Aus.  16216729. 

24.  719x729=whiitr 

25.  92730465xl794  =  wh.it  >  Aus.  1G635845421U. 
2(5.  81G2035x28r>45=whatr'  Au3.  233801492575. 


DIVISION  OF   ABSTRACT  NUMBERS. 


§  41.  Division  is  the  operation  of  finding  how  many 
times  one  number  is  contained  in  an  otlier.  Thus,  4  in  20, 
">  times  :  her«  20  is  divided  hy  4,  since  4  is  contained  5 
times  in  20. 

§  42.  Or,  Division  is  the  operation  of  separating  a  num- 
ber into  some  number  of  equal  parts.  Thus,  if  20  is  di- 
vided into  4  equal  parts,  each  or  the  parts  is  5. 

§  4$.  The  number  to  he  divided  is  called  the  dividend. 

i^  44.  The  dividing  number  is  called  the  diviHor. 

§  45.  The  result  of  division  is  called  the  quotievl. 

§  46.  When  tlie  division  is  not  complete,  the  undivided 
p^rt  of  the  dividend  is  called  the  remainder.  Thus,  Sin  35, 
4  times,  with  3  over  ;  here  35  is  the  dividend,  8  is  the  divi- 
sor, 4  is  the  quotient,  and  3  is  the  remainder. 

§  47.  The  sign  of  division,  -T-,is  read  dividcdhy.  When 
placed  between  twu  numbers,  it  denotes  that  the  one  before 
it  is  to  be  divided  l>y  tlie  one  after  it.     Thus,  20-7-5::rr:4. 

§  48.  Division  is  sometimes  denoted  by  placing  the  divi- 
dend over  the  divisor  with  a  line  between  them.  Tiius, 
?i-=r4. 

9r, 


Z') 


^48 


ABSTRACT    NUMBERS. 


I3I^V^lS10I«ir    TJLBLE. 


1  in 

0, 

no  time     | 

2  in     0, 

no  time 

3  ] 

n 

0, 

no  time 

1  in 

1, 

onee 

2  in     2, 

once 

o 

n 

13, 

Onee 

3    in 

2, 

twice 

2  in     4, 

twice 

3  ] 

m 

6, 

twice 

1  in 

3,' 

3  times 

2  in     6, 

3  times 

3  ] 

n 

0, 

3  times 

1  in 

4, 

4  times 

2  in     8, 

4  times 

3  i 

n 

12, 

4  times 

1  iu 

5, 

5  times 

.     2  in  10, 

5  times 

3 

in 

15, 

5  times 

1  in 

^, 

6  times 

2  in   12, 

6  times 

3 

n 

18, 

6  times 

1   in 

7, 

7  times 

2  in  14, 

7  times 

3  ] 

n 

21, 

7  times 

I  in 

8, 

8  times 

2  in  10. 

8  times 

3 

LU 

24, 

8  times 

1  in 

9, 

9  times 

2  in  18, 

9  times 

3 

m 

27, 

9  time^ 

4  in 

X 

no  time 

^  5  in     0, 

no  time 

6  ] 

m 

"^"o, 

no  time 

4-in 

4, 

once 

5  in     5, 

once 

6 

m 

6, 

once 

4  in 

8. 

twice 

5  in  10, 

twice 

6 

m 

12, 

twice 

4  in 

12, 

3  times 

5  in  15, 

3  times 

6 

in 

18, 

3  times 

4  in 

16, 

4  times 

"^5  in  20, 

4  times 

6 

m 

24, 

4  times 

4  in 

20, 

5  times 

5  iu  25, 

5  times 

6 

n 

30, 

5  times'. 

4  in 

24, 

6  times 

5  iu  30, 

6  times 

6 

m 

36, 

6  times 

4  in 

26, 

7  times 

5  in  35, 

7  times 

6 

n 

42, 

7  times 

4  iu 

32, 

8  times 

5  in  40, 

8  times 

6 

n 

48, 

8  times 

4  iu 

36, 

9  times 

5  in  45, 

9  times 

6  : 

LU 

54, 

9  times 

7  in 

0, 

no  time 

8  in     0, 

no  time 

9  : 

n 

0, 

no  time 

7  in 

7, 

once 

8  in     8, 

once 

,9 

in 

9, 

once 

7  in 

14, 

twice 

8  in  16, 

twice 

9 

m 

18, 

twice 

7  in 

21, 

3  times 

8  in  24, 

3  times 

9 

m 

27, 

3  times 

7  in 

28, 

4  times 

8  in  32, 

4  times 

9 

U 

36, 

4  times 

7  in 

35, 

5  times 

8  in  40, 

5  times 

9 

m 

45, 

5  times 

7  in 

42, 

6^  times 

8  in,  48, 

6  timo-: 

9 

m 

54, 

6  times 

7  in 

4VJ, 

7  times 

8  in  56, 

7  times 

9 

in 

63, 

7  times 

7  in 

66, 

8  times 

8  in  64, 

8  times 

9 

in 

72, 

8  times 

7  in 

63, 

9  (times 

8  in  72, 

9  times 

9 

in 

81, 

9  times 

10  in 

0,^ 

no  time 

11  in     0, 

no  time 

'12" 

in 

i. 

),  no  time 

10  in 

10, 

once 

11  in  11, 

once 

12 

in 

V. 

I,  once 

10  in 

20, 

twice 

11  in  22 

twice 

12 

in 

2- 

:,   twice 

10  in 

30, 

3  times 

11  in  83, 

3  times 

12 

n 

86 

),  8  times 

10  in 

40, 

4  times 

11  in  44 

4  times 

12 

m 

4^ 

\,  4  times 

10  in 

50, 

5  times 

11  in  55 

5  times 

12 

in 

6( 

),  5  times 

10  in 

60, 

6  times 

11  in  66, 

5  times 

12 

m 

71 

J,  6  times 

10  in 

70, 

7  times 

11  in  77 

7  times 

12 

Q 

8^ 

t,  7  times 

10  in 

80, 

8  times 

11  ia  88 

'  8  times 

12 

m 

9( 

J,  8  times 

10  in 

90. 

9  tiinos 

11   iri  99 

9  timp= 

12 

in 

10^ 

5,  9  times 

26 


DIVISION   OF   INTEGERS-  §50 


I.   SHORT  DIVISION. 

Ex.  1.  Divide  3096  by  3. 

J)W^<^r,r^  3)3096,  Dividend.  ^  49^  Model.— 3  in  3, 

1032,  Quotient.  once  ;  3  in  0,  no  time  ;  3  in 

9,  3  times;  3  in  6,  twice. 
The  quotient  is  1032. 

Explanation. — The  divisor  is  placed  on  the  left  of  the 
liviilotid.  Then,  beginning  at  the  left,  the  number  in  each 
rder  of  units  is  divided  by  3,  and  each  quotient  figure  is 
et  in  its  proper  column. 

Ex.  2.  Divide  806X4  by  2.  Quot.  40312. 

3.  Divide  8048  by  4.  Quot.     2012. 

4.  Divide  90369  by  3.  Quot.  30123. 
5..  Divide  17120  by  8. 

8)17120  g  50    MopEL.— 8  in  17,  twice,  with  I  over, 

2140         set' down  2  ;  8  in  11,  once,  with  3  over,  set 
down  1  ;   8  in  32,  4  times ;  8  in  0,  no  time. 
The  quotient  is  2140. 

Explanation. — 8  is  not  contained  in  1,  that  is,  in  1  ten- 
thousand,  in  its  present  form  ;  hence,  1  ten-thousand  is  re- 
duced to  10  thousands,  and  added  to  the  7  thousands,  mak- 
ing 17  thousands.  8  is  contained  twice  in  16  ;  so  that  there 
i3  1  thousand  still  undivided.  This  is  reduced  to  10  hun- 
dreds, and  added  to  the  1  hundred,  making  11  hundreds. 
8  is  contained  once  in  8  ;  so  that  there  are  3  hundreds  still 
undivided.  These  are  reduced  to  30  tens,  and  added  to 
the  2  tens,  making  32  tens.  8  is  contained  in  32  just  4 
times.  The  0  of  the  dividend  is  retained  in  the  quotient, 
to  cause  the  several  quotient  figures,  2  thousands,  1  hun- 
dred, and  4  tens,  to  occupy  their  proper  plares 

.27 


§51  ABSTRACT   NUMBERS. 

Ex.  6.  Divide  36374  by  9. 

9)36374  §  51^    Model.— 9  in  36,  4  times;  9  in 

4041... 5      3,  0  time,  with  3  over,  set   down   0  ;    9  in 

37,  4  times',  with  1  over,  set  down  4  ;  9  in 

14,  once,  with  5  over,  set  down  1  in  the  quotient,  and  5  a? 

remainder.     The  quotient  is  4041,  and  the  remainder  5. 

Explanation. — The  division  of  the  5  units  might  be 
denoted  ^-,  as  in  §  48. 

Rule. — -Set  the  divisor  on  the  left  of  the  dividend,  unth  a 
line  between  them,  and  one  under  the  dividend. 

Beginning  at  the  left,  see  how  many  times  the  divisor  is 
roiitained  in  each  figure  of  the  dividend,  and  set  the  result 
u/nder  the  dividend. 

Whenever  there  is  a  remainder,  prefix  it  to  the  next  fgure 
of  the  dividend,  before  dividing. 

If  the  divisor  is  not  contained  in  any  figure,  except  tlw 
first^  set  0  under  such  figure,  and  regard  it  as  a  remainder, 

pjROOF. — Multiply  the  quotient  by  the  divisor  :  the  prod-' 
uct,  increased  by  the  remainder,  if  any,  will  be  equal  to 
th©  dividend. 

Ex.  7.  Divide  73052  by  2.  Quot.  36526. 

8.  Divide  222345  by  3.  Quot.  74115. 

9.  Divide  123456  by  4. 

lO!  Divide  790530  by  5.  Quot.  158106, 

11.  Divide  78920472  by  C.  Quot.  13153412. 

12.  Divide  945  by  7.    * 

13.  Divide  1240128  by  8.      .  Quot.  155016. 

14.  Divide  743200173*^  by  9.  Quot.  82577797. 

15.  Divide  4703750  by  10. 

16.  Divide  9009  by  11.  Quot.  819. 
i7.  Divide  721428  bv  12.  Quot.  60119. 

28 


DIVISION    OP    INTEGERE.  §52 


U.  LONG  DIVISION. 


Ex.  18.  Divide  2966232  by  925. 

Dividend,  29662321925,  Divisor. 

2775^  13206,  Quotient  ,  .,    ^,  ^  .  ■ 

ig^Q  29,  3  times  ;  multiply 

.  _-^ .  the    divisor    by    3  ;    3' 

^7"^'^  times   5    are    15,    set 

'^:^j^  down  5  ;  3  times  2  are 

682,  Remainder.         6,  and  1  are  7  ;  3  times 

9  are  27,  5et  down  27  : 
ubtract  the  product  from  the  dividend;  2;  5  from  G 
leaves  1  ;  7  from  16  leaves  9;  8  from  9  leaves  1  : — 9  in 
19,  twice;  multiply  the  divisor  by  2;  twice  5  are  10,  set 
down  0;  twice  2  arc  4,  and  1  are  5;  twice  9  are  18,  set 
down  18  :  subtract  the  product  from  the  previous  remain- 
der ;  3 ;  U  from  2  leaves  2 ;  5  fi^om  11  leaves  6;  9  from  9 
leaves  0: — 9  in  6,  no  time ;  annex  2: — 9  in  62,  6  times; 
Multiply  the  divisor  by  6  ;  6  times  5  are  30,  set  down  0  ; 
()  times  2  are  12,  and  3  arc  15,  set  down  5;  6  times  9  are 
54,  and  1  are  55  :  subtract  the  product  from  the  previous 
remainder ;  0  from  2  leaves  2  ;  5  from  13  leaves  8  ;  6  from 
12  leaves  6;  6  from  6  leaves  0.  The  quotient  iso206,nnd 
the  remainder  682. 

Explanation. — The  divisor  is  placed  on  the  right  of  the 
dividend,  for  convenience  in  multiplying.  The  number  0 
is  used  throughout  as  a  trial  divhor.  As  two  figures  ©f  the 
real  divisor  arc  thus  omitted,  two  figures  of  each  partial 
dividend  must  be  omitted  also.  Hence,  in  the  third  step, 
we  say  9  in  6,  and  not  9  in  62,  until  we  have  annexed  an 
additional  figure.  The  first  quotient  figuile  stands  for 
3000  ;  hence  the  first  product  is  really  2775000,  and  the 
first  remainder  191232;  but,  as  we  do  not  need    all   these 

29 


>53  ABSTRACT    NUMBERS. 


figures  for  tke  next  step,  we  begin  to  subtract  only  one 
place  to  the  right  of  the  last  valuable  figure  in  the  prod- 
uct- The  division  of  the  remainder  might  be  expressed 
as  in  §  48. 

Ex.  19.  Divide  6593  by  19. 

6593119 

57      oT«  §  53.  Model. — 2  in  6,  3  times;  multiply 

~7."q~-  the  divisor  by  3  ;    3  times  9  are  27 ;  sot 

nn.  down  7  ;  o  times  1  are  3,  and  2  are  5  :  sub- 

tract  the  product   from  the   dividend  ;  9  ; 

^^^  7  from  15  leaves  8;  6  from  6  leaves  0  :-^ — 

^'^^  2  in  8, 4  times  ;  multiply  the  divisor  by  4  ; 

0  4  times  9  are  36,  set  down  6  ;  4  times  1  are 

4,  and  3  are  7  :  subtract  this  product  from 

the  previous  remainder ;  3  ;  6  from  9  leaves  3 ;  7  from  8 

leaves  1  : — 2  in  13,  7  times;  multiply  the  divisor  by  7 ;  7 

times  9  are  63,  set  down  3  ;  7  times  1  are  7,  and  6  are  13, 

set  down  13  :  subtract  the  product  from  the  previous  re- 

Qiainder;   0.     The  quotient  is  347. 

Explanation. — If  the  second  figure  of  the  divisor  is 
less  than  5,  the  first  figure  is  the  trial  divisor ;  but,  if  the 
second  figure  is  greater  than  5,  the  trial  divisor  is  one  more 
than  the  first  figure.  If,  on  multiplying,  a  quotient  figure 
be  found  to  be  too  large  or  too  small,  let  it  be  diminished  or 
increased  a  unit  at  a  time  until  the  right  result  is  attained. 

Rule. — Set  the  divisor  on  the  right  of  the  dividend,  with 
a  line  between  ihem,  and  one  under  the  divisor. 

Beginning  at  the  left,  see  how  often  the  divisor  is  contained 
in  the  first  part  of  the  dividend :  the  result  will  he  the  first 
figure  of  the  quotient.  Multiply  the  divisor  by  this  quotient 
figure,  and  subtract  the  product  from  that  part  of  the  divi- 
dend which  loas  used,  annexing   to  the  remainder  the  next 

figure  of  the  dividend. 

30 


DIVISION    OF    INTEGERS.  §53 


Take  this  remainder  as  a  second  partial  dividend,  and 
from  it  obtain  the  second  quotient  Jif/ure.  Muhiphj  (he  divi- 
sor  by  thi^Jigure,  and  subtract  the  product  from  the  previous 
remainder^  annexing  to  the  second  remainder  the  next  figure 
of  the  dividend. 

Continue  this  process  till  all  the  figures  of  ili'-  dividend 
have  been  used. 

If  any  partial  dividend  will  iiot  contain  the  divisor,  set  0 
in  the  quotient^  annex  an  other  figure  of  the  dividend,  and 
divide  again. 

Proof.  1. — The  same  as  in  §  51,  for  short  division. 

Or,  2.  Subtract  the  remainder,  if  any,  from   the  divi 
dend  ;  divide  this  remainder  by  the  quotient,  and  the   re- 
sult will  be  the  divisor. 

Ex.  20    Divide  18950  by  25.  «  Quot.  758. 

21.  Divide  17136  by  36. 

22.  Divide  42581  by  49  Quot.  8t)9. 

23.  Divideud=lG7'01,  Div-i.-ur^57  Quot.  293. 

P^  Note. — Begin,    "Divide  the  Dividend  by  tbo  Divuor.'' 

24.  Dividend— 265oG,  Divi3or^()2. 

25.  Dividend^l5076872,  Divisor=:72.     Quot.  209401. 

26.  Dividend— 30744,  Divisor=:84.  Quot.  366. 

27.  Divisor=97,  Dividend==84002. 

"  28.  Divisor=^125,  Dividend-=15625.  Quot.  125. 

29.  Divisor=:273,  Dividendr=104832.  Quot.  384. 

30.  Divisor=354,  Dividend=94l64. 

31.  Divisor=465,  Dividend^2G7375.  Quot.  575. 

32.  Divisor=531,  Dividend=340902.  Quot.  642. 

33.  Divisor=^685,  Dividend=543205. 

34.  Divisor:==721,  Dividend=r2728264.  Quot.  3784. 

35.  DiTisor=829,  Dividend=5697717. 

31 


.^63  ABSTRACT    NUMBERS. 


36.  r>ivisor=937,  Dividendr=981976.  Qu»t.  1048. 

37.  5754375-4-1125=:whatl  Ans.  5116. 

38.  4515625-^2125=:what! 

39.  48284964-T-3094=what  ?  Ans.  15606. 

40.  24896825~-4105=what  ?  Ans.  6065. 

41.  27206656-T-5216=what? 

42.  45782172-^6327=what?  Ans.  7236. 

43.  313201258-f-7153=what?  Ans.  43T86. 

44.  293834463995  ~  8405==what  f 

45.  572473044-i-9516=what?  Ans.  60159. 

46.  93939874943-- 10471=what?  Ans.  8971433. 

47.  151807041— 12321==what? 

48.  Dividend=l 2741 53376,  Divisor=23456. 

Qwot.  54321. 

49.  Dividend==18p9739176,  I>iTisor=34056. 

Quot.  54Q2I. 

50.  Dividend=2642079580,  Divisor=40565. 

51.  Dividend==:2900 124304,  Divisorzz=56504. 

Quot.  513p. 

52.  Divisorrr=:65405,Dmdend=6677 19645. 

Quot.  1020». 

53.  Divisoi— 74316,  Dividend=4734969624. 

54.  DiTisor=81634,  Dividend=7571 145330. 

Quot.  92745. 

55.  Dmsor=:95703,  Dividend=1299551037. 

Quot.  13579. 

56.  Divisor=97531,  Dividend=2999956029. 
.57.  Divisoi— 36805,  l)ividend==800655970. 

Quot.  21754, 

58.  Divisor=::234282,  Dividend=83596737522. 

Quot.  356821. 

59.  Divisor=5276431,  DiTidend^7105901 538475. 

32 


CONTRACTED   ADDITION   OF  INTEGERS. 


§54 


CONTRACTION   IN  ADDITION. 


;  Note. — The  judicious  teacher  will  omit  this  and  most  of  the 
following  contractions  as  his  classes  proceed  through  the  book  the 
first  time. 

Ex.  1.  Add  together  the  following  numbers  : 


469375 
237924 
472437 
853214 
975318 
242326 


§54.  Model.— 26  and  10  are  36,  and  8 
are  44,  and  10  are  54,  and  4  are  58,  and  30 
are  88,  and  7  are  95,  and  20  are  115,  and  4 
are  119,  and  70  are  189,  and  5  are  194,  set 
down  94: — 1  and  23  are  24,  and  50  are  74, 

and  3  are  77,  and  30  are  107,  and  2  are  109, 

3250594  and  20  are  129,  and  4  arc  133,  and  70  are 
203,  and  9  are  212,  and  90  are  302,  and  3 
are  305,  set  down  05  : — 3  and  24  are  27,  and  90  are  117, 
and  7  are  124,  and  80  are  204,  and  5  are  209,  and  40  are 
249,  and  7  are  256,  and  20  are  276,  and  3  are  279,  and  40 
are  319,  and  6  are  325,  set  down  325.     The  sum  is  3250594. 

Explanation. — Beginning  at  the  right,  and  taking  two 
columns  at  a  time,  we  take  in  first  the  tens  and  then  the 
units,  as  we  go  up  the  column,  and  set  down  the  two  right 
hand  figures  of  each  sum. 


Ex.  2. 

123456 
789012 
345678 
901234 
567890 
987654 
821098 
765432 

4801454 


o. 

1234 

5678 
9012 
3456 
7890 
1357 
9246 
8987 

46860 


4. 

235689 
124578 
135792 
468097 
531086 
420987 
654321 
555775 


5. 

14250663 
32215941 
10340285 
92341967 
82395786 
17084657 
40558476 
91623378 


6. 

819349 
720258 
630167 
541076 
452985 
363894 
274703 
185612 


7. 

120341 
989052 
878163 
767274 
656385 
545496 
432107 
321098 


3126325  380811153 


38 


§55  ABSTRACT    NUMBERS. 


CONTRACTION  IN  SUBTRACTION. 


Ex.  1.  From  970347  take   the   sum   of   14375;   226899, 
12534,  and  369708. 

?Z?!?^  §  55.  Model.— 8  and  4  are  \2,  and  9  are 

14375         21,  and  5  are  26,  from  27  leaves  1  ;  2  and  3 

226899  are  5,  and  9  are  14,  and  7  are  21,  from  24 
12534         leaves  3  ;  2  and  7  are  9,  and  5  are  14,  and  8 

^69708         are  22,  and  3  are  25,  from  33  leaves  8  ;  3  and 

346831  9  are  12,  and  2  are  14,  and  6  are  20,  and  4 
are  24,  from  30  leaves  6  ;  3  and  6  are  9,  and 

1  are  10,  and  2  are  12,  and  1  are  13,  from  17  leaves  4  ;  1 
and  3  are  4,  and  2  are  6,  from  9  leaves  3.  The  remainder 
is  346831. 

Explanation. — As  26,  the  sum  of  the  subtrahend  units, 
can  not  be  take^  from  7,  the  units  of  the  minuend,  we  add 

2  tens,  that  is,  20  units,  to  the  minuend,  and  afterwards 
add  2  tens  to  the  subtrahend.     (§  28.) 

yote. — Let  the  pupil  be  required  to  use  this  contraction  when- 
ever it  can  be  applied. 

Ex.  2.  From  1000  take  9+98-f-176-f  254-f  289. 

Rem.  174. 

3.  From  9125  take  8-f  88 +888 -f  1297+3945. 

Rem.  2899. 

4.  From  10275  take  1245 +  373 5 +  298 6 J- 895. 

Rem.  1414. 

5.  From  87579  take  1477+2796  +  8972  +  10896. 

Rem.  63438. 

6.  From  120225  take  246+1357+97531  +  1358. 

7.  From  72575  take  575+2575+4575+15575. 

8.  From  4970  take  250-|-325-|-348-|-2211. 

9.  From  22907  take  3916.|-2821-|-4302-|-2309. 

34 


CONTRACTED    MULTIPLICATION  OF  INTEGERS. 


§57 


CONTRACTIONS  IN  MULTIPLICATION. 


Ex.  1.  Multiply  7325  by  100. 

752500  ^  ^^'  ^^^^^i" — Annex  two  naughts  to  the 

multiplicand.     The  product  is  732500. 

Explanation. — We  annex  to  the  multiplicand  as  many  ci- 
phers as  there  are  annexed  to  the  1  of  the  multiplier.  (§  39.) 
Ex.  2.  Multiply  1358  by  10.  Prod.  13580. 


3.  Mult 

4.  Mult 

5.  Mult 
5.  Mult 
7.  Mult 


ply  2468  by  100. 
ply  4579  by  1000. 
ply  86725  by  10000. 
ply  1020  by  100. 
ply  32500  by  1000. 


Prod.  246800. 
Prod.  4579000. 


H.  Multiply  32500  by  25000. 
32500 


__25000 

1625 

650 

812500000 


§  57.  Model.— 5  times  5  are  25,  set 
down  5  ;  5  times  2  are  10,  and  2  are  12, 
set  down  2  ;  5  times  3  are  15,  and  1  are 
16,  set  down  16: — twice  5  are  10,  set 
down  0  under  2 ;  twice  2  are  4,  and  1 
are  5  ;  twice  3  are  6  :— add  the  partial 
products  :  5  ;  2  ;  5  and  6  are  11,  set  down  1  ;  1  and  6  are  7, 
and  1  are  8  :— annex  5  naughts.  The  product  is  81250000o! 
Explanation. — After  finding  the  product  of  the  valua- 
ble figures,  we  annex  to  it  as  many  naughts  as  there  are  in 
the  right  of  both  the  factors. 

Ex.  9.  Multiply  27500  by  350. 

10.  Multiply  1250  by  1500. 

11.  Multiply  747000  by  250. 

12.  Multiply  19500  by  1400. 

13.  Multiply  124750  by  3000. 

14.  Multiply  2795000  by  2700. 

85 


Prod.  9625000. 

Prod.  1875000. 

Prod.  186750000. 

Prod.  27300000. 


§58  ABSTRACT   NtJMBERS, 


15.  Multiply  3759  by  104. 

3759x104  I  58^  Model.— 4  times  9  are   36,   set 

_j_^^*^ down  6,  two  places  to  the   right   of   9  ;  4 

390936     '        times  5  are  20,  and  3  are  23,  set  down  3  ; 

4  times  7  are  28,  and  2  are  30,  set  down 

0 ;  4  times  3  are  12,  and  3  are  15,  set  down  15  : — add  the 

partial  products : — 6  ;  3  ;  9 ;  5  and  5  are  10,  set    down    0  ; 

1  and  i  are  2,  and  7  are  9  5  3.     The  product  is  390936. 

Explanation.— If  the  multiplier  has  only  two  valuable 
figures,  the  first  of  which  is  1,  we  multiply  by  the  other 
valuable  figure,  and  set  the  first  figure  of  the  product  as  far 
to  the  right  of  the  units  figure  of  the  multiplicand  as  this 
figure  is  to  the  right  of  the  1 . 

Ex.  16.  Multiply  2376  by  12.  Prod.  28512. 

17.  Multiply  47475  by  107.  Prod.  5079825. 

18.  Multiply  57875  by  10080.  Prod.  583380000. 

19.  Multiply  275  by  1009.  Prod.  277475. 

20.  Multiply  4479  by  10006. 

21.  Multiply  795310  by  10500. 

22.  Multiply  1025  by  7001. 

1025x7001  §  59.  Model.— 7  times  5  are    35, 

7175  set  down  5,  three  places  to  the  left  of 

7176025  5  ;  7  times  2  are  14,  and  3  are  17,  set 

down  7 ;  7  times  0  are  0,  and  1  is  1  5 

7  times  1  are  7  : — add  the  partial  products  : — 5  ;  2  ;  0  ;  5 

and  1  are  6  ;  7  ;  1  ;  7.     The  product  is  7176025. 

Explanation. — If  the  multiplier  has  only  two  valuable 
figures,  the  last  of  which  is  1,  we  multiply  by  the  other 
valuable  figure,  and  set  the  first  figure  of  the  product  as 
far  to  the  left  of  the  units  figure  of  the  multiplicand  as  this 
figure  is  to  the  left  of  the  1. 

Ex  23.  Multiply  7893  by  51.  Prod.  402543. 

36 


CONTRACTED    MULTIPLICATION  OF  INTEGERS. 


m 


24.  Multiply  4685  by  601. 

25.  Multiply  23795  by  7010. 

26.  Multiply  1375  by  8001. 

27.  Multiply  20478  by  90010, 


Prod.  2815685. 
Prod.  166802950. 


28.  Multiply  27346  by  99. 

"'^o-^^fi  ^  ^^'  ^^^^^T^-— ^^^^ex  2  naughts    to    the 

.J*l___         multiplicand  : — subtract    the    multiplicand 

2707254         from  the  result;  6  from  10  leaves  4;  5  from 

10  leaves  5  ;  4  from  6  leaves  2 ;  7  from  14 

leaves  7  ;  3  from  3  leaves  0 ;  0  from  7  leaves  7  ;  0  from  2 

leaves  2.     The  product  is  2707254. 

Explanation. — Since  9  is  1  less  than  10,  we  may  multi- 
ply any  number  by  9,  by  subtracting  the  number  from  10 
times  itself.  If  therefore  the  multiplier  consists  of  9's 
alone,  we  annex  to  the  multiplicand  as  many  naughts  as 
there  are  nines  in  the  multiplier,  and  subtract  the  multi- 
plicand from  the  result. 

Ex.  29.   iVIultiply  124795  by  9. 

30.  Multiply  24735  by  99. 

31.  Multiply  1469  by  999. 

lultiply  70095  by  99. 
53.  Multiply  9999  by  256.  (§  37.) 

34.  Multiply  1276538  by  999. 

35.  ^Multiply  8365712  by  99. 


Prod.  1123155. 
Prod.  2448765. 
Prod.  1467531. 
Prod.  6939405. 
Prod.  2559744. 


36.  Multiply  2754  by  54. 

27540  on-.     ,r  r.     ^    . 

§61.  Model. — o4=9  times  6.  First,  mul- 
tiply by  9  :—  (§  60.)  4  from  10  leaves  6  ;  6 
from  14  leaves  8  ;  8  from  15  leaves  7 ;  3  from 
7  leaves  4  ;  0  from  2  leaves  2.  The  product 
is  24786.  Multiply  this  product  by  6  :—  6 
times  6  are  36,  set  down  6  ;  6  times  8  are  48, 
and  3  are  51,  set  down  1  ;  6  times  7  are  42,  and  5  are  47, 

37 


2754 

24f86 
6 

148716 


m 


ABSTRACT    NUMHKRS. 


set  down  7 ;  6  tiroes  4  are  24,  and  4  arc  28,  set  down  8 ;  6 
times  2  are  12,  and  2  are  14,  set  down  14.  The  product 
is  148716. 

Explanation. — If  the  multiplier  is  the  product  of  two 

or  more  numbers,  we  may  multiply  the  multiplicand  by 

either  of  those  numbers,  and  this  product  by  an  other,  and 

so  on. 

Ex.  37.  Multiply  3725  by  35. 

38.  Multiply  17075  by  48. 

39.  Multiply  473729  by  49. 

40.  Multiply  279o6  by  56. 

41.  Multiply  124684  by  64. 

42.  Multiply  247372  by  72. 


Prod.  130375. 

Prod.  843G00. 

Prod.  23212721. 

Prod.  1564416. 


43.  Multiply  21857  by  714. 
21857 


714 

149499" 
298998 

T5248898 


§62.  Model. — 14  is  twice  7.  First, 
multiply  by  7  : —  7  times  7  arc  49,  set  down 
9  under  7  of  the  multiplier  ;  7  times  o  are 
35,  and  4  are  39,  set  down  9 ;  7  times  3 
are  21,  and  3  are  24,  set  down  4  ;  7  times 
1  are  7,  and  2  are  9  ;  7  times  2  are  14,  set 
down  14.  The  product  is  149499.  Multiply  this  product 
by  2  : —  twice  9  are  18,  set  down  8  under  4  of  the  multi- 
plier; twice  9  are  18,  and  1  are  19,  set  down  9;  twice  4 
are  8,  and  1  are  9  ;  twice  9  arc  18,  set  down  8  ;  twice  4  are 
8,  and  1  are  9  ;  twice  1  are  2.  Add  the  partial  pr(-ducts  : 
8  ;  9  ;  9  and  9  arc  1>^,  set  down  8  ;  1  and  ^^  arc  9,  and  9  are 
18,  set  down  8  ;  1  and  9  are  10,  and  4  are  14,  set  down  4  ; 
1  and  2  are  3,  ami  9  are  12,  set  down  2  ;  1  and  4  :n\.  5  ;  1. 
The  product  is  15248898. 

Explanation. — If  one  part  of  the  multiplier  is  a  factor 
of  an  other,  the  work  may  be  contracted  as  in  the  model, 
placing  the  first  figure  of  each  product  immediately  under 

88 


CONTRACTED  MULTIPLICATION  OP  INTEGERS. 


§64 


the  right  hand  figure  of  the  corresponding  part  of  the  mul- 


tiplier. 

Ex.  44.  Multiply  12479  by  654. 

45.  Multiply  24793  by  56248. 

46.  Multiply  97635  by  53545. 

47.  Multiply  86436  by  497. 

48.  Multiply  23047  by  488. 

49.  Multiply  902756  by  366108, 


Prod.  8161266. 

Prod.  1394556664. 

Prod.  5227866075. 

Prod.  42958692. 


50.  Multiply  225  by  25. 

4)22500  ^(53^    Model.— Annex  2  naughts  to  the 

5625         multiplicand  : — divide  the  r-esult  by  4  : — 4 

in  22,  5  times,  with  2  over,  set  down   5 ;  4 

in  25,  6  times,  with  1  over,  set  down   6;    4  in    10,  twice, 

with  2  over,  set  down  2 ;  4  in  20,  5  times.     The  product  is 

5625. 

Explanation. — Annexing  2  naughts  multiplies  by  100, 
(§  56)  :  hence,  since  100=4x25,  we  divide  the  product  by 
4,  to  get  the  true  product. 

Ex.  51.  Multiply  10275  by  25.  Prod.    256875. 

52.  Multiply  28832  by  25.  Prod.    720800. 

53.  Multiply  72725  by  25.  .             Prod.  1818125. 

54.  Multiply  84287  by  25.  Prod.  2107175. 

55.  Multiply  96248  by  25. 
5G.  Multiply  8324728  by  25. 

57.  Multiply  274  by  125. 

8)274000  ^  (54    Model.— Annex  3  naughts  to  the 

34250         multiplicand  : — divide  the  result  by  8  : — 8 

in  27,  3  times,  with  3  over,  set  down   3  ;  8 

in  34,  4  times  with  2  over,  set  down  4 ;  8  in  20,  twice,  with 

4  over,  set  down  2  ;  8  in  40,  5  times  ;  8  in  0,  no  time.  The 

product  is  34250. 

39 


§65  AHBTRACT    MMBBRS. 


Explanation. — Annexing  3  naughts  multiplies  by  1000, 
(§  56):  hence,  since  1000—8x125,  wc  divide  the  product 
by  8,  to  get  the  true  product. 

Kx.  58.  Multiply  125  Ly  125.  Prod.  15620. 

59.  Multiply  625  by  125.  Prod.  78125. 

60.  Multiply  1776  by  125.  Prod.  222000. 

61.  Multiply  34070  by  125.  Prod.  4259875. 

62.  Multiply  934478  by  125. 

63.  Multiply  7840349  by  125. 


CONTRACTIONS  LN  Di\  iiilO.X. 

Ex.   1.  Divide  l25wl 

125  64  ^^  ^^    MoDKL. — Cut  off  two  figures  at   thf 

'  right.     The  quotient  is  125,  and  the  remain 

der  64. 

ExPL-\NATiON. — We  cub  off  at  the  right,  for  remainder, 
a.s  many  figures  as  there  are  naughts  at  the  right  of  the  I 
•  •f  the  divisor.  The  remaining  figures  on  the  leftconstitut« 
the  quotient. 

2.  Divide  34000  by  10.  (]uot.  3400. 

3.  Divide  74500  by  100.  Quot.  745. 

4.  Divide  UJ740  by  100.  Quot.    107;  Hem.  40. 

5.  Divide  24G000  by  lOUO.  Quot.  24(;. 
0.  Divide  147375  by  1000. 

7.  Divide  24680  by  100. 

8.  Divide  98630  by  800. 

8,00)086,30  ^QQ  MoDKL.^Out  off  the  2naught.s 

123 — 230         at  the  right  of  the  divisor,  and  2  fig- 
ures at  the  right  of  the  dividend  :  — 
then,  8  in  0,  once,  with  1  over,  set  down  1  ;  8  in  18,  twice, 
with  2  over,  set  down  2  ;  8  in  2(.),  3  times,  with  2  over.  Tho 
(quotient  is  123,  and  th(?  rcmaindo^-  '^'^'^ 

40 


CONTRACTED   DIVISION    OF   INTEGERS.  §68 


Explanation. — The  remainder  after  dividing  is  prefixed 
to  the  dividend  figures  cut  off,  to  constitute  the  true  re- 
mainder. 

Ex.  9.  Divide  127569  by  270.      (.^uot.  4724;  Rem.  189. 

10.  Divide  56000  by  700.  Quot.  80. 

11.  Divide  3230000  by  1700.  Quot.  190U. 

12.  Divide  24600  by  2400.  Quot.  10  :  Hem.  600. 

13.  Divide  7346790  by  72900. 

14.  Divide  135073  by  21800. 

15.  Divide  275  by  5. 

^"^[^   '  §  67.  Model.— Multiply  the  dividend  by  2  : 

_^         twice  5  are  10,  set  down  0  ;  tv/ice  7  are  14,  and 

55,0         1  are  15,  set  down  5  ;  twice  2  are  4,  and  1    are 
5  : — divide  this  product  by    10.     (§  65.)     The 
(quotient  is  55. 

Explanation. — Since  the  dividend  is  already  5  times 
the  required  quotient,  multiplying  it  by  2  gives  (2x5)  10 
times  the  quotient.  The  part  cut  off  at  the  right,  by  this 
plan,  is  twice  the  true  remainder. 

Ex.  16.  Divide  10024  by  5.  Quot.  2004;  Kern.  4. 

17.  Divide  2725  by  5.  Quot.  545. 

18.  Divide  49720  by  5.  Quot.  9944. 

19.  Divide  598405  by  5.  Quot.  119681. 

20.  Divide  479324  by  5. 

21.  Divide  2379156  by  5. 

22.  Divide  329  by  25. 

^2^  i^  68.  Model.— Multiply  the  dividend  by  4  . 

4  times  9  =  30,  set  down  6 ;  4  times  2=8,  and  3 

i    /i-  =ll,set  down  1;  4times3  =  12, and  1  =  13, set 

down  13  : — divide  this  product  by  100    (J;  65.) 

The  quotient  is  13,  and  the  remainder  4. 

41 


§69  ABSTRACT    NUMBERS. 

Explanation. — Since  the  dividend  is  already  25  times 
the  required  quotient,  multiplying  it  by  4  gives  (4x25) 
100  times  the  quotient.  The  part  cut  off  at  the  right,  by 
this  plan,  is  4  times  the  true  remainder. 

Ex.  23.  Divide  293235  by  25.      Quot.  11729;  Rem.  10. 

24.  Divide  148532  by  25.  Quot.  5941 ;  Rem.  7. 

25.  Divide  2475  by  25.  Quot.  99. 
.  26.  Divide  193450  by  25.                                 Quot.  7738. 

27.  Divide  34795  by  25. 

28.  Divide  107059  by  25. 

29.  Divide  23725  by  125. 

23725  I  Qg    Model. — Multiply  the  dividend  by 

^         8:8  times  5  are  40,  set  down  0  ;  8  times  2 


189,800         are  16,  and  4  are  20,  set  down  0  ;  8  times  7 

are  56,  and  2  are  58,  set  down  8  ;  8  times  3 

are  24,  and  5  are  29,  set  down  9  ;  8  times  2  are  16,  and    2 

are  18,  set  down  18  : — divide  this  product  by  1000.  (§65.) 

The  quotient  is  189,  and  the  remainder  100. 

Explanation. — Since  the  dividend  is  already  125  times 
the  required  quotient,  multiplying  it  by  8  gives  (8x125) 
1000  times  the  quotient.  The  part  cut  off  at  the  right,  by 
this  plan,  is  8  times  the  true  remainder. 

Ex.  30.  Divide  724350  by  125.  Quot.  5794;  Rem.  100. 

31.  Divide  111000  by  125.  Quot.  888. 

32.  Divide  246625  by  125.  Quot.  1973. 

33.  Divide  57935  by  125.  Quot.  463  ;  Rem.  60. 

34.  Divide  793575  by  125. 

35.  Divide  125364  by  125. 

36.  Divide  10202  by  42. 

42 


CONTRACTED   DIVISION    OP   INTEGERS.  §71 


2)10202 


3)5101 


§  70.  Model. — 42=2  times  3  times  7. 

First,  divide  by  2  : — 2  in    10,    5    times; 

7)1700—1         2  in  2,  once  ;  2  in  0,  no  time  ;  2   in   2, 

242 — 6         once  : — divide  this  quotient  by  3  : — 3  in 

5,  once,  with  2  over,  set  down   1  ;  3    in 

21,  7  times;  3  in  0,  no  time  ;  3  in  1,  no  time,  with  1  over, 

set  down  0  in  the  quotient,  and  1    as   remainder  : — divide 

this  quotient  by  7 : — 7  in  17,  twice,  with  3  over,  set    down 

2  ;  7  in  30,  4  times,  with  2  over,  set  down  4 ;  7  in  20,  twice, 

with  G  over,  set  down  2  in  the  quotient,  and  6  as  remainder. 

The  quotient  is  242,  and  the  remainder  38. 

Explanation. — If  the  divisor  is  the  product  of  two  or 
more  numbers,  we  may  divide  the  dividend  by  either  of 
those  numbers,  and  the  quotient  by  an  other,  and  so  on.  The 
true  remainder  is  found  by  multiplying  each  remainder  by 
all  the  divisors  previous  to  the  one  which  produced  it,  and 
adding  together  the  several  products. 

Ex.  37.  Divide  7346  by  56.  Quot.     131 ;  Hem.  10. 

38.  Divide  347934  by  35.  Quot.  9940  ;  Rem.  34. 

39.  Divide  92384  by  64.  Quot.  1443;  Rem.  32. 

40.  Divide  83495  by  45.  Quot.  1855  ;  Rem.  20. 

41.  Divide  745106  by  72. 

42.  Divide  656215  by  96. 

43.  Divide  34635  by  285. 

^1^^^|^_?5  §  71.  Model.— 3  in  3,  once  :— once  5  is 

J^r  121         5,  from  6  leaves  1;  once  8   is  8,  from  14 
;*^^  leaves  6  ;  once  2  is  2,  and  1  are  3,  from  3 

^^^  leaves  0  :  annex  3  : — 3  in  6,  twice  : — twice 

5  are  10,  from  13  leaves  3  ;  twice  8  are  16, 
and  1  are  17,  from  21  leaves  4  ;  twice  2  are  4,  and  2  are  6, 
from  6  leaves  0  :  annex  5  : — 3  in  4,  once  :— -once  5  is  5,  from 
5  leaves  0;  once  8  is  8,  from  13  leaves  5  ;  once  2  is  2,  and 
1  arc  3,  from  4  leaves  1.  The  quotient  is  121,  and  the  re- 
mainder 150. 

43 


§72  ABSTRACT    NUMBERS. 


Explanation. — The  products  are  not  written,  but  are 

iui mediately  substracted  as  in  §  55. 

Note. — Let  all  the  exercises  in  Long  Division  hereafter  be   per- 
formed by  this  plan. 

Ex.  44.  Divide  136895  by  725.  Qiiot.  188  ;  Rem.  595. 

45.  Divide  247986  by  836.  Quot.  296  ;  Rem.  230. 

46.  Divide  358097  by  749.  Quot.  478  ;  Rem.    75. 

47.  Divide  469108  by  5275.  Quot.  88  ;  Rem.  4908. 

48.  Divide  5702195  by  4386. 
40.  Divide  68132050  by  5295. 


GENERAL  PRINCIPLES  OF  DIVISION. 


§  72.  If  the  divisor  remain  unchanged,  and  the  dividend 
be  multiplied  by  any  number,the  quotient  will  be  multiplied 
by  the  same  number.     Thus,   32-i-8=4  :    then,  64-^8=8. 

§  73.  If  the  divisor  remain  unchanged,  and  the  dividend 
be  divided  by  any  number,  the  quotient  will  be  divided  by 
the  same  number.     Thus,  32-f-8=4  :  then,  16-^8  =  2. 

§  74.  If  the  dividend  remain  unchanged,  and  the  divisor 
be  multiplied  by  any  number,  the  quotient  will  be  divided 
by  the  same  number.     Thus,  32-^8=4  :  then,  32-^16  =  2. 

§  75.  If  the  dividend  remain  unchanged,  and  the  divisor 
be  divided  by  any  number,  the  quotient  will  be  multiplied 
by  the  .-ame  number.     Thus,  32^8=4:    then,  32-^4=8. 

§  76.  If  the  dividend  and  the  divisor  be  both  multiplied 
by  the  same  number,  the  quotient  will  remain  unchanged. 
Thus,  32-8.-4:  then,  64-^16:zr=4. 

§  77.  If  the  dividend  and  the  divisdr  be  both  divided  by 
the  same  number,  the  quotient  will  remain  unchanged. — 
Thus,  32-T-8==l:  then,  lG--4=ri. 

44 


PROMISCUOUS   PROBLEMS.  5J80 


PROMISCTTOTTS    PROBLEMS. 


1.  The  subtrahend  is  thirty  thoiisaDcl  and  forty-five ;  the 
remainder  is  fortj'-six  thousand  eight  hundred  and  ninety: 
what  is  the  minuend  ?  Ans.  769i:j5. 

§  78.  Minuend — Subtrahend=Remainder. 
Minuend — Remainder=Subtrahend. 
Subtrahend  4- Remainder.-rrrMinuend. 

2.  The  minuend  is  three  hundred  thousand ;  the  subtra- 
hend is  ninety-nine  thousand  three  hundred  and  seventy- 
four :  what  is  the  remainder?  Ans.  200024. 

3.  The  minuend  is  seventy  thousand  and  twenty-nine  ; 
the  remainder  is  sixty-five  thousand  and  forty-six  :  what  i.* 
the  subtrahend  ? 

4.  The  multiplicand  is  twenty-seven  thousand  and  four  ; 
the  product  is  seven  hundred  and  twenty-nine  millions,  two 
hundred  and  sixteen  thousand,  and  sixteen  :  what  is  the 
multiplier  ?  Ans.  27004. 

§  79.  Multiplicand  X  Multiplier =Product. 
Product-7-Multiplier= Multiplicand. 
Product-f- Multiplicand  =  Multiplier. 

5.  The  multiplicand  is  four  thousand  and  seventy-two  : 
the  multiplier  is  one  thousand  one  hundred  and  six :  what 
is  the  product  1  Ans.  4503632. 

6.  The  product  is  ninety-three  thousand  three  hundred 
and  sixty-one  ;  the  multiplier  is  eighty-nine  :  what  is  the 
multiplicand  1 

7.  The  divisor  is  one  thousand  and  nine  ;  the  quotient  is 
nine  hundred  and  ten  :  what  is  the  dividend  ?   Ans.  918190. 

§  80.  Dividend-i-Divisor=Quotient. 
Divisor  xQuotient=Dividend. 
(Dividend — Remainder) -^Quotient =Divisor. 
Quotient  xDivisor-f- Remainder =Dividend. 
45 


§)^0  ABSTRACT    NUMBERS. 


8.  The  dividend  is  nineliuudred  and  forty- five  thousand, 
eight  hundred  and  eighty-eight ;  the  divisor  is  two  thousand 
and  four  :  what  is  the  quotient  ?  Ans.  472. 

9.  The  dividend  is  one  hundred  and  forty-eight  thou- 
sand; the  quotient  is  three  hundred  and  forty-two;  the  re- 
mainder is  two  hundred  and  fifty-six  :  what  is  the  divisor  ? 

10.  The  quotient  is  one  thousand  and  three  ;  the  divisor 
is  one  thousand  and  two  :  the  remainder  is  one  thousand 
and  one  :  what  is  the  dividend  ?  Ans.  1006007. 

11.  Find  the  sum  of  two  hundred  and  forty-five  thou- 
sand, nine  hundred  and  seven,  seventy-four  thousand  and 
seventy-four,  one  hundred  and  nine  thousand  and  nine,  and 
three  hundred  and  ninety-seven.  Sum,  429387. 

12.  Find  the  difference  between  two  hundred  thousand, 
and  one  hundred  and  eighty-seven  thousand  six  hundred 
and  fifty-four. 

13.  Find  the  product  of  one  million  three  hundred  and 
seventy-five,  and  one  thousand  three  hundred  and  seventy- 
five.  Prod.  1375515625. 

14.  Find  the  quotient  of  three  millions  divided  by  six 
thousand  two  hundred  and  seventy-nine. 

Quot.  477 ;  Rem.  4917. 

15.  What  number  is  that  from  which  if  2407,  4072, 724, 
and  7240  be  subtracted,  the  remainder  will  be  7042  ? 

16.  What  number  is  that  to  which  if  2407,  4072,  724, 
and  7240  be  added,  the  sum  will  be  15000  ?  Ans.  557. 

17.  What  number  is  that  by  which  if  2047  be  multiplied, 
the  product  will  be  15151894  1  Ans.  7402. 

18.  What  number  is  that  by  which  if  2025042  be  di- 
vided, the  quotient  will  be  2021  ? 

19.  247-f  1023— 9344-3720— 4142-f245=:what? 

20.  (247-154)-^3-f(247-f  154)x3=what?  Ans.  1234. 

46 


PROMISCUOUS  PROBLEMS,  §81 


§  81.  A  parenthesis  enclosing  two  or  more  numbers  shows 
that  their  united  value  is  to  be  subjected  to  the  operation 
indicated  immediately  before  or  after  the  parenthesis.  For 
example,  in  the  preceding  problem,  the  difference  of  247 
and  154  is  to  be  divided  by  3,  and  the  sum  of  247  and  154 
is  to  be  multiplied  by  3,  and  the  product  and  the  quotient 
arc  to  be  added  together. 

Two  numbers  thus  connected  are  called  a  binomial ^  three 
numbers  are  called  a  trinomial;  four,  a  tetranomial ;  five. 
a  pcntanomial ;  six,  a  hexanomial,  &c. 

The  20th  problem  is  read,  '^  Binomial  247  minus  154 
divided  by  3  plus  binomial  247  plus  154  multiplied  by  3  is 
equal  to  what?" 

21.  3247  +  247-47  +  7-(247— 474-7)=what? 

22.  (987— 876-f333)-H(765-543)-f-210-95=what  y 

Ans.  117. 

23.  27— 30-^10-f  (475  — 399)H-4=:what?  Ans.  43. 

24.  (204-60)-^6-(90-|.10)--5-[-(76-!.12)-T-4=what  '^ 

25.  204— 60-f-6— 90-l-10-^5-K76-|-12)--4:=what  ? 

Ans.  128. 

26.  (204— G0)-^6-j-90-l-10-5-|-(76-|-12)^4r=:what  ? 

27.  (204— 60)--6— (90-|-10)-f-5-|-7G— 12-^4r=what  ? 

28.  204— 60--6— (90-|-10--5.!-76)— 12-^4=rwhat  ? 

Ans.  23. 

29.  123-1-41— (123— 41>|-123x41—123^41=what? 

Ans.  5122. 

30.  123-1-41— (123— 4l)-|-(123x41— 123) ^41==what  ? 

31.  J[(742-v-2)-r-53]x27-lJ-^53=::what?  Ans.  1. 

32.  [(199-78)-f-ll-(199-43)--78]x(12-3)=lwhat? 

Ans.  81. 

33.  [(117-43)  x2]--37-K138-128)  x37==^hat  1 

47 


ABSTRACT    NUMBERS. 


MEASURES  AND  MULTIPLES. 


§  82,  An  eve7i  number  is  one  which  can  be  exactly  divided 
by  2.     Thus,  12,  4,  36,  58,  and  70,  are  even  numbers. 
Xote. — All  even  numbers  end  in  either  2,  4,  6,  8,  or  0. 

§  So.  An  odd  number  is  one  which  can  not  be  exactly  di- 
vided by  2,     Thus,  9,  17,  25,  33,  and  41,  are  odd  numbers. 
Kote, — AH  odd  numbers  end  in  either  1,  3,  5,  7,  or  9. 

§  84.  A  prime  number  is  one  which  is  not  the  product  of 
two  other  numbers.  Thus,  2,  3,  5,  7,  11, 13,  17, 19, 23, 29, 
31,  37,  41,  43,  47,  53,  59,  61,  67,  71,  73,  79,  83,89,  and  97, 
are  all  the  prime  numbers  less  than  100. 

X-A^BLK    of   I*I?,I^i:]K   INTUMCBEl^^S   ixp    to    lOOO. 


o 

43 

103  1 

173 

241 

317 

401 

479 

571 

647 

739 

827 

1 
919 

3 

47 

107  1 

179 

251 

331 

409 

487 

577 

653 

743 

829 

929 

0 

53 

109 

181 

257 

337 

419 

491 

687 

659 

751 

839 

9,37 

.  1 

59 

113' 

191 

263 

347 

421 

499 

593 

661 

757 

853 

941  i 

lll 

Gl 

127 

193 

269 

349 

431 

603 

599 

673 

761 

857 

947! 

jl8 

67 

131 

197 

271 

353 

433 

509 

601 

677 

769 

859 

963 

!l7 

71 

137 

199 

277 

359 

439 

521 

607 

683 

773 

863 

967 

119 

73 

139 

211 

281 

367 

443 

523 

613 

691 

787 

877 

971 

'23 

79 

149 

223 

283 

373 

449 

511 

617 

7U1 

797 

881 

977 

29 

83 

151 

227 

293 

379 

457 

547 

619 

709 

809 

883 

983 

31 

89 

157 

229 

307 

383 

461 

557 

631 

719 

811 

887 

991 

37 

97 

163 

233 

311 

389 

463 

563 

641 

727 

821 

907 

997 

41 

101 

167 

239 

313 

397 

467 

569 

643 

733 

823 

911 

§  85.  A  composite  number  is  one  which  is  the  product  of 
two  other  numbers.  Thus,  4,  6,  9,  15,  21,  and  30,  are 
composite  numbers,  because  2x2=4,  2x3=6,  3x3=9, 
3x5=15,  3x7=21,  and  5x6=30. 

Is  20  prime,  or  composite  ?  25  ?  28  ?  31  ?  34?  37  ?  40  '/ 
43?  501  57?  64?  71?  78?  85?  92?  99?  106?  217? 
328  ?   439  ? 

48 


MEASURES  AND  MULTIPLES.  §87 

§86.  Powers. — The  first  jyo we r  of  a  number  is  the  num- 
ber itself.  Thus,  5  is  the  Jirst  power  of  5 ;  7,  of  7 ;  10,  of  10. 

The  second  poicer  of  a  number  is  the  product  of  the  num- 
ber multiplied  by  itself.  Thus,  8G  is  the  second  power  of 
G,  because  6x6=^36:  81,  of  9,  because  9x0—81  :  100,  of 
10,  because  10x10=100. 

The  second  power  of  a  number  is  usually  called  its  square. 

The  third  power  of  a  number  is  the  product  of  the  num- 
ber multiplied  by  its  square.  Thus,  8  is  the  third  power  oi 
2,  because  2x4=^8  :  64,  of  4,  because  4x16=64  :  216,  of 
6,  because  6x36=216:  512,  of  8,  because  8x64=--:5l2: 
1000,  of  10,  because  10x100  =  1000. 

The  third  power  of  a  number  is  usually  called  its  cube. 
In  like  manner,  what  is  the  fourth  power  of  a  number  ? 
What  is  the  sixth  power  't     The  7iinth  power  ?     ka. 

§  87.  Roots. — The  first  root  of  a  number  is  the  number 
itself. 

The  second  root,  or  the  square  root,  of  a  number  is  one  of 
the  tico  equal  factors  which  produce  it.  Thus,  5  is  the  square 
root  of  25,  because  5x5=25. 

15  has  no  square  root,  because  its  two  factors,  3  and  5,  are 
not  equal. 

The  third  root,  or  the  cube  root,  of  a  number  is  one  of  the 
three  equal  factors  which  produce  it.  Thus,  3  is  the  cube 
root  of  27,  because  3.3.3=27. 

30  has  no  cube  root,  because  its  three  factors,  2,  3,  and  5, 
are  not  equal.  25  has  none,  because  it  has  only  two  equal 
factors,  5  and  5.  16  has  none,  because  it  has  four  e«^ual 
factors,  2,  2,  2,  and  2. 

In  like  manner,  what  is  the  fourth   root  of  a  number  ? 
What  is  the  seventh  root  ?     The  sixteenth  root  ?     &c. 
D  49 


§88  ABSTRACT    KUiMIiEKS. 

•^  bS.  Th.Q  prime  f (I do i's  oi  n  composite  nuitiber  arc  the 
prime  numbers  by  whose  continued  multiplication  the  num- 
ber is  produced.  Thus,  the  prime  factora  of  9  are  3  and  3/ 
because  3x3=9  :  the  prime  factors  of  GO  are  2,  2,  3,  and 
5,  because  2.2.3.5=60. 

§  89.  A  measure  of  a  number  is  a  number  which  is  con- 
tiuned  in  it  a  number  of  times  without  a  remainder.  Thus, 
3  is  a  measure  of  12,  because  3  is  contained  exactly  4  time.s 
iri  12 :  4  is  a  measure  of  30,  because  4  is  confeaiiied  exacfli/ 
9  times  in  30. 

Is  5  a  measure  of  10  ?  25  ?  37  ?  40  ?  63  ?  Q>~j  \  80  ? 

Is  6  a  measure  of  7  ?  12?  20?  30?  39?  48?  54? 

Is  7  a  measure  of  14?  19?  28?  36?  42?  48?  63? 

§  90.  A  multiple  of  a  number  is  a  number  which  containn 
it  a  number  of  times  loithout  a  remainder.  Thus,  12  is  a 
multiple  of  3,  because  12  contains  3  exactly  4, times  :  36  is 
x  multiple  of  4,  because  36  contains  4  exactly  9  times. 

Is  40  a  multiple  of  2  ?  3  ?  4  ?,  5  ?  6?  7?  8?  9?  10?  20? 

Is  56  a  multiple  of  2  ?  4  ?  7  ?  8  ?  9  ?  10  ?  14  ?  20  ?  24  ? 

Is  60  a  multiple  of  2?  3?  4?  5?  6?  7?  8?  10?  12?  15? 

§91.  (7o?nmo?i  means  belonging  equally  to  two  or  more 
numbers. 

§  92.  One  number  is  a  common  measure  of  two  or  more 
numbers,  if  it  is  a  measure  of  each  of  them.  Thus,  3  is  a 
measure  of  9,  also  of  12,  also  of  18  ;  hence,  3  is  a  common 
measure  of  9,  12,  and  18.  Also,  4  is  a  common  measure  of 
8,  24,  32,  ai)d  48. 

Is  2  a  common  measure  of  4,  6,  and  10  ? 

Is  3  a  common  measure  of  6,  10,  and  15  ? 

Is  4  a  common  measure  of  12,  16,  and  20  ? 

Two  or  more  numbers  may  have  several  common  meas- 

50 


iires.     Tlui.,  :, .id  361iave  as  c.......i....  ......,.^i'cs  2,  o,  4, 

6,  and  1*2.     In  this  case,  lU  is,  of  course,  the  greatest  com- 
DKjn  ::!easure  of  2-i  and  36. 

§  93.  Ohe  number  is  a  common  multiple. of  two  or  more 
numbers,  if  it-  is  a  multiple  of  each  of  them.  Thus,  40.  is  a 
multiple  of  5,  also  of  8,  also  of  10  ;  hence,  40  is  a  common 
uiultiple  of  5,  8,  and  10.  Also,  45  i-  a  o.ovir.i.nu  multiple 
of  3,  5,  and  9. 

Ls  10  a  common  multiple  of  2  and  5  ? 
Is  15  a  common  multiple  of  3  and  6  ? 
Is  oO  a  common  multiple  of  2,  5,  and  10  .' 

Two  or  more  numbers  alwajs  have  several  common  mul- 
tiples. Thus,  4,  3,  and  6,  have  as  common  multiples  12, 
34,  3t3,  48,  60,  &o.  In  this  case,  12  is,  of  course,  the  kcial 
eommon  multiple  of  4,  3,  and  6. 

§  94.  Two  or  more  numbers  are  prime  to  each  other,  if 
they  have. np  common  measure.  Thus,  81  and  64  are  prime 
to  each  other.     Also,  20,  27,  and  77  are  prime  to  each  other. 

§  95.  2  is  a  measure  of  every  number  which  ends  in  ei- 
ther 2,  4,  6,  8,  or  0.     (§  82.  Note.) 

o  is  a  measure  of  a  number,  if  it  is  a  measure  of  the  sum 
of  the  figures  which* denote  the  number.  ThuS;  3  is  a  meas- 
ure of  246,  or  462,  or  624,  or  612,  or  426,  or  261,  or  2064, 
or  4602,  &c.,  because  3  is  a  measure  of  6-f  4-f  2,  that  is. 
of  12. 

4  is  a  measure  of  a  number,  if  it  is  a  measure  of  the  num- 
ber denoted  by  its  two  right  hand  figures.  Thus,  4  is  a 
measure  of  768,  or  1860,  or  95372,  or  1112316,  because  4 
is  a  measure  of  68,  or  60,  or  72,  or  16. 

5  is  a  measure  of  every  number  which  ends  in  either  5 

or  0.     Thus,  5  is  a  measure  of  20,  or  55,  or  100,  or  275. 

51 


J{96  AESTE.ACT    NUMBERS. 


o  is  a  measure  of  erery  even  number  of  which  3  .is  a 
measnre.  Thus,  6  is  a  measure  of  462^  or  4512,  or  1236  : 
but  not  of  471,  or  632L 

8  is  a  measure  of  a  number,  if  it  is  a  measure  of  the 
number  denoted  by  its  three  right  hand  figures.  Thus,  8 
is  a  measure  of  34800,  or  753064,  because  8  is  a  measnre  of 
800,  or  64. 

9  is  a  measure  of  a  number,  if  it  is  a  measure  of  the  sum 
of  the  figures  which  denote  the  number.  Thus,  9  is  a  meas- 
ure of  891,  or  1728,  or  253269,  because  9  is  a  meavsure  of 
18,  or  18,  or  27. 

10  is  a  measure  of  every  number  which  ends  iii  0. 

100  is  a  measure  of  every  number  which  ends  in  2  naughts. 
Is  2  a  measure  of  3040  ?  4047?  28?  1112?  10124? 
Is  3  a  measure  of  258?  869 ?  12345678 ?  5169  ?  2571 '( 
Is  4  a  measure  of  125784  ?  24680  ?  57932  ?  14760  ?  1 1 12? 
Is  5  a  measure  of  245?  12450?  7824?  12570?  3457? 
Is  6  a  measure  of  570  ?  378  ?  45S42  ?  123456  ?  12324  ? 
Is  8  a  measure  of  5070120  ?  247080?  1479008?  1234? 
Is  9  a  measure  of  1234566  ?  68472  ?  1357  ?  1476  ? 
Is  10  a  measure  of  240  ?  245  ?  3795  ?  7630  ?  1460  ? 

§  96.  A  measure  of  a  number  is  a  measure  of  any  one  of 
its  multiples.  Thus,  6  is  a  measure  of  18:  then  it  is  a 
measure  of  36,  or  54,  or  72,  or  90. 

§  97.  A  common  measure  of  two  or  more  numbers  is  a 
measure  of  their  sum.  Thus,  8  is  a  common  measure  of 
16,  24,  and  40  :  then  it  is  a  measure  of  80. 

§  98»  A.  common  measure  of  tAO  numbers  is  a  measure  of 
their  difference.     Thus,  9  is  a  common  measure  of  18  and 
.^4  :  then  it  is  a  measure  of  36. 

52 


PRIME    FACTORS.  §99 


PulME    FACTORS. 


Ex.  1.  Resolve  7'*^00  into  its  priDie  factors. 

2)7800 

2)3900  ^^^-  J^I^r>EL. — Divide   the  namber  by  2. 

'MI^Qi^  (§50).     Divide  tlie  quotient  by  2.     Divide 

•^-C  this  cjuotieut  by  2.     Divide  this  quotient  by 

'^1?.I5  5.     Divide  this  quotient  by  5.     Divide  this 

5)195  quotient  hj  3.     This  quotient  is  a  prime  uum- 

jj^gT^  ber.     The  prime  f-icfnrs  of  7800  arc  2,  2,  2, 

'  1.  '_         5,  5,  3,  and  l.S. 

o 

Explanation. — It  is  better  to  divide  first  by  2  as  often 
as  possiblcj  then  b}'-  5,  and  then  by  the  other  prime  numbers 
in  succession.  The  several  divisors  and  the  last  quotient 
are  evidently  the  prime  factors, of  the  number. 

Rule. — Divide  the  given  niiviber  by   one   of  it^  prime 
nimsures ;  divide  the  quotient  hy  one  of  its  prime  measures  ; 
continue  thus  dividing  until  a  ^^ rime  number  is  obtained  fo 
a  quotient:  the  several  divisor.^  and  the  last  quotient  will  be 
ihe  prime  factors  of  the  given  nu,'7\ber. 

Proof. — The  continued  product  of  the  prime  factors  will 
he  equal  to  the  given  number. 

Ex.  2.  Resolve  524  into  its  prime  factors. 

P.  F.  2,  2,  and  131. 

3.  Resolve  460  into  its  prime  factors 

4.  Resolve  770  into  its  prime  factors. 

P.  F.  2,  5,  7,  and  11, 

5.  Resolve  880  into  its  prime  factors. 

P.  F.  2,  2,  2,  2,  5,  and  11. 

6.  Resolve  999  into  its  prime  factors. 

7.  Find  the  prime  factors  of  1040. 

P.  F.  2,  2,  2,  2,  5,  and  13, 
53 


§100  ABSTRACT    NTJMBTJRS. 

8.  Find  the  prime  factors  of  1160. 

P.  F.  2,  2,  5,  and  29. 

9.  Find  the  prime  factors  of  1275. 

iO.  What  are  the  prime  factors  of  1300  t 

Ans.  2,  2,  l),  5,  and  13. 
iL  What  are  the  prime  factors  of  1590  ? 

An&.  2,  5,  3,  and  53. 

12.  What  are  the  prime  factors  of  1738? 

13.  What  are  the  prime  factors  of  19500  ? 

Ans.  2,  2,  5,  f),  5,  3,  and  15. 

14.  What  are  the  prime  factors  of  966000  ? 

Ars  2,  2,  2,  2,  5,  5,  5,  3,  7,  and  23. 
15    What  are  the  prime  factors  of  825000  ? 
16.  What  arc  the  prime  factors  of  13572001 

Ans.  2,  2,  2,  2,  5,  5,  3.  3,  13,  and  29. 


INVOLUTION. 

§  100.  Involution  is  the  process  of  finding  i^  power  oi  -^ 
number.  From  the  definitions  of  the  several  powers  in 
§86,  it  is  evident  that  any  power  of  a  number  is  fo raid  L_j 
taking  the  number  as  a  factor  in  miiltiplication  as  nianj 
times  as  there  are  units  in  the  number  of  the  power, 

Ex.  1.  What  is  the  square  of  7?  Ans   49, 

2.  What  is  the  cube  of  3  ?  Ans.  27. 

3.  Yv^hat  is  the  fourth  power  of  2  't 

4.  What  is  the  fifth  power  of  2  ?'  Ans.  32. 

5.  What  is  the  fourth  power  of  5  ?  Ans   625. 

6.  What  is  the  cube  of  9  ? 

7.  What  is  the  square  of  19?  Ans.  361. 

8.  What  is  the  cube  of  15  ?  Ans.  3375. 

9.  What  is  the  fourth  power  of  20? 

54 


JiVOLUTJON.  §102 


EV(^LUT10X. 


^\0l  EvoLU^i*iON  is  the  process  of  finding,  a  >oo^  of  ;i 
given  power.  The  method  here  explained  is  applicabli 
only  to  such  numbers  as  have  precise  roots.  The  method 
of  extracting  aj»proxiraate  roots  of  imperfect  powers  can 
not  be  explained  without  the  use  of  algebraic  formulas,  and 
consequently  is  not  given  in  this  treatise. 

Ex    1.  What  is  the  cube  root  of  21G  ? 

r 2)2 16 


2)108 

2^54 


§  102.  MoDEii.-— Resolve  the  given  nuRi- 
ber  into  its  prime  factors.    (§99.)     It  con- 
po)'-7         tains  three  twos  and   three  threes.     Hence. 
3y  >=  cube  root  is  2x8=6. 


Explanation. — Since  the  cube  root  of  a  number  is  ouv; 
of  the  three  equal  factors  which  produce  it,  we  separate  the 
prime  factors  into  sets  of  three  equal  prime  factors  each, 
and  selecting  one  from  each  setjthe  produc^  of  those  ^elected 
is  evidently  the  cube  roqt  of  the  given  nilmber.  For  ari  v 
other  root,  wo  separate  into  sets  of  as  many  prime  factors 
each  a?  there  ape  units  in  the  ri-.mber  of  the  root. 

3'  he  prime  factors  ')e*eparattd  as   above,  '.ho 

required  rv)ot  can  not  be  exactly  found,  eitlun-  by  this,  or  by  afly  vtfK'r 
method. 

Kui.         .-■   .    ..:/.;.    .,.:;:.„  ^  ■...,..-  into  ila  .....,::  fac!or>> : 

separate  the  factors  into  groups  of  as  many  equal  factor,'. 

each  as  there  are  units  in  the  niimher  of  the  root}  select  one 

faetor  from  each  f^reiip,  and  multiply  togetlter  (hose  selected  : 

their  product  tcill  he  the  root  n  quired. 

Proof  — Raise  the  root  to  the  corresponding  power.  Tht^ 
result  will  bo  counl  to  the  given  number. 

55 


§10o  ABSTRACT    NUMBERS. 


GREATEST  COMMOX  MEASURE. 


Ans. 

6. 

An». 

3, 

Ans. 

10. 

Ans. 

4. 

Ex.  2.  What  is  the  square  root  of  100  ?  Ans.  10. 

3.  What  is  the  cube  root  of  125  ? 

4.  What  is  the  fourth  root  of  1296  ? 

5.  What  is  the  fifth  root  of  243  ? 

6.  What  is  the  sixth  root  of  64  ? 

7.  What  is  the  fourth  root  of  10000  1 

8.  What  is  the  fifth  root  of  1024  ? 

9.  What  is  the  cube  root  of  3375  ? 

10.  What  is  the  square  root  of  12321  ?  Ans.  111. 

11.  What  is  the  square  root  of  65536  ?  A^a^.  256, 

12.  What  is  the  fourth  root  of  65536  ? 

13.  What  is  the  eighili  .■  .,t  of  65536? 

14.  What  is  the  sixteenth  root  of  65536  ? 

15.  What  is  the  square  root  of  390625  ? 

16.  What  is  the  fourth  root  of  390625?" 

17.  What  is  the  eighth  root  of  390625  ? 

18.  Yfhat  is  the  cube  root  of  10077696? 

19.  What  is  the  ninth  root  <'.f  10077696? 

20.  What  is  the  cube  root  oi  42875  ? 

21.  What  is  the  square  root  of  122500  ? 

22.  What  is  the  square  root  of  7569? 


Ans. 

4. 

Aus. 

t.) 

Ans. 

25. 

Ans. 

5. 

Ans. 

6. 

Ans. 

35. 

Ans. 

87„ 

Ex.  1, 
and  480. 

2)60, 

.  Ein 
150, 

;d  the 
480 

5)30, 

75, 

240 

3)  6, 

15, 

48 

2, 

5, 

16 

2. 

,5.3=oU 

grep.test  common    measure  of  60,   150, 

§  103.  Model  — Divide  each  of  the 
given  numbers  by  2.  (^-  50).  Divide 
each  of  these  quotients  bj  5.  Divide 
each  of  these  quotients  bji  3.  These 
quotients  are  prime  to  each  other. 
2.5.3=30.  30  is  the  greatest  com- 
mon measure  of  the  2:iven  numbers. 
56 


GREATEST    COMMON    MEASURE. 


Explain A'iiuN. — In  this  operaiiuu  it  u  uul  iicoe^haij  n>i 
the  divisors  to  be  prime  nuiubcTs.  Wo  might  liave  divided 
by  10  and  by  3,  or  by  5  and  by  6. 

lluLE. — Divide  each  of  the  given  numbers  hy  any  one  cj 
their  common  meamrcs ;  divide  each  of  these  qnoiienta  by 
ne  of  iheir  comm07i  measures ;  continue  f hits  diriding  un- 
,  il  the  quotientc,  become  prime  to  each  other :  the  continued 
product  of  the  divisors  will  be  the  grcated  comrnnii.  mt'.asurc 
q  f  the  g iven  n k m bers . 

Ex.  li.  Fin'"'  t'"' 'greatest  common  measure  of  '^6,  12'), 
-16,  and  234. 

30=2.2.3.3  §  104.  Modkl.— Resolve  36 

126  =  2.    3.3.7  inj-Q  ity  pi-iine  factors.    (§  99). 

'?16  =  2.2.3.3.    2.3  36=2.2.3.3.    Resolve  126  into 

234=2^_3.3. 13  ^^^  primefactors.  126=2.3.3.7. 

2.3.3.  =  18.  Resolve  216  into  its  prime  fac- 

tors. 216=2.2.3.3.2.3.  Re- 
solve 234  into  its  prime  factors.  234=2.3.3.13.  2.3.3  =  18. 
18  is  the  greatest  common  measure  of  the  ?iven   nnm];ei"». 

ExPLANATlo.v. — The  prime  factors  are  arranged  Vr-ith 
equal  factors  in  the  same  column,  as  far  as  possible.  The 
full  columns  contain  the  factors  that  are  common  to  aU  the 
numbers.  The  product  of  these  factors  is  the  greatest 
common  mea.sure  of  tlio  numbers. 

Rule. — Resolve  each  of  the  given  numbers  info  il.s  pri/iw 
factors ;  select  those  factors  ichich  are  common  to  all  the 
numbers:  the  continued  product  of  these  factors  will   l-r   iJtc 

(jreatest  common  measure  of  the  given  numbers. 

57 


§105  ABSTRACT   NUMBERS. 


o.   Find  the  greatest  common  measure  of  108  and  261 
261:108 


216 


2 


108!45 
90!^~-  ■^*  105.  Model.— Divide  261  by  108. 

il  (S  71).     Divide  108  by  45.     Divide  45 

f^^^  hy'lii.     Divide  18  by  9.     There  is  do 

'-''^  2  reoiainder,     9  is  the  greatest   common 

>o7^ —  measure  of  the  (nven  numbers. 

_____ 

Explanation. — 0  is  a  measure  of  IS,  (§  89);  hence  it  i.^ 
a  measure  of  2x18,  or  36,  (§  96) ;  hence,  of  86-f  9,  or  45, 
(§  97) ;  hence,  of  2  x45,  or  90,  (§  96)  ;  hence,  of  108,  {%  hi)  : 
hence,  of  216,  (§96);  hence,  of  261,  (§  97);.. hence  it  is  a 
common  measure  of  108  and  261. 

YhULE^—r-DLvldeihe  larger  ?iumber  h;j  the    smalUr ;  then 
divide  the  smaller  numher  by  the  remainder^    and  continue 
divvUnq  the  last  divinor  bv  the  laM.  remainder. until  there  ?*? 
no  reinuinder  :  tlie  last  divisor  iriU  he  fitc   r/reatcst   co'mrn^ 
rr.  :'■    ■  ,'  •    of  the  given  numhers. 

To  jind  the' r/rentest  common  'me<!^ure  of  mor*'.  than  two 
numbers^  find  the  greatest  common  measure  of  two  of  them, 
(hen  find  the  greatest  co7nnio?i  measure  of  this  mmmire  and 
an  other  of  the  mnnbos^  and  w  on  :  th'  last  common  meas- 
ure will  he  Ihe  greateU  common  measwe  of  alt  the  numher; . 

Either  of  the  above  methods  may  be  used  in  the  follow- 
ing exercises. 

Ex.  4.  Find  the  greatest  common  measure  of  48,  64,iipd 

H2.  '  G.  0.  M..d«. 

5.  Find  X\iq  greatest  common  measure  of  08,  1 19,  and  357. 

(I.   Find  the  greatest  conimon  messuro  of  GO.  OjQ,  and  10'-. 


GREATEST   COUMON    MEASUr..K.  >^i()5 


7.   Vi   d  the  greatest  common  measure  of  c  : 

1  .:;.  (1.  C.  M.  13. 

i!^    '       '    lie  greatest  commou  mcasjuro  of  40,  60,  and  200. 

G.  C.  M.  2U. 
9.  Find  the  greatest  common  measure  of  96,  lt?8,  and  320, 

10.  Fiud  the  greatest  common  measure  o^  164,  '-87> 
and  451.                                                        ^  0,  M.  41. 

11.  Find  the  greatest  common  mea«iirc  of  63, 126,  i^  15,  and 
441.  '  G.  0.U,  68. 

12.  Find  the  greatest  common  measure  of  150,  '376,  and 
(i75. 

13.  Find  the  greatest  common  measure  of.  40,  CO,  68,  and 
204.  ^^  ^\  M.  4. 

H.   FiM.^  0-  .      '   :;  r  2,   nnfl 

n:)-..  M..214. 

I  :\   Find  the  greatest  common  measure  of  63. 189, 315,  and 

}i}.  Fuid  the  greatefat  common  me.'i^u re  of  152,  *j80,  and 
532,  ,.      ..^-.  '  ■  r.    r'.  jy[,  7^. 

17.  Fijid  the  greatest  common  mcasv.  ■■  1    ,       7,   and 

'^^IL          ^  G.  C.  M.  17. 

i  '.  d  the  greatest  comuion  measure  of  iOQ,'  IJ^O,  210. 
and  4JV~. 

19.  Find  til  i34,    190,    and 
•     1140.  G.  CM.  38. 

20.  Find  the  greate.'ic  common  measure  of   54,    108,  324, 

nnd378.  G   C.  M.  54. 

Si.  Find  the  greatest  common  measure  of  oQ,  84,  140,  and 

106.  *  a.  C.  M.  28. 

22.   Find  the  greatest  common   Rinr.sHro  of  75,    1:25,   375, 

and  67^.  G    C.  M.  75. 

Fin<]  the  greateist  common  measure  of  46, 115,  aad  161 . 

50 


106  ABSTRACT    NUMBERS. 


LEAST  COMMON  MULTIPLE. 


2)40, 

60,  150 

2)  JO, 

30,    75 

^3)10, 

15,    75 

3)2, 

:J,     15 

2. 

1.       5 

Ex.  1.   Find  the  least  common  multiple  of  40,  60,and  150. 

§  J  00  Model. — Divide  each  of 
the  numbers  by  2.  (§  50).  Divide 
some  of  the  quotients  by  2.  Divide 
each  of  these  quotients  by  5.  Divide 
some  of  these  quotients  by  3.  These 
quotients  are  prime  to  each  other. 

__ 2.2.5.3.2.5=600.     600  is  the  least 

2.2.5.3.2.5=600         common  multiple  of  the  given  num-^ 
bers. 

Explanation. — We  divide  two  or  more  of  the  given 
numbers  by  any  prime  number  that  will  divide  them  with- 
out a  remainder  ;  and  two  or  more  of  the  resulting  num- 
bers by  any  prime  number  that  will  divide  them  without  a 
remainder :  and  so  on,  till  the  quotients  are  prime  to  each 
other : — remembering  to  repeat  in  the  line  below,  such 
numbers  as  cannot  be  divided.  By  this  means,  every  factor 
of  each  number  is  used,  and  hence  the  result  is  a  commov 
multiple  of  the  numbers;  but  no  factor  of  either  number  is 
used  more  than  once,  and  hence  the  result  is  their  least 
common  multiple. 

p^uij/. — Divide  two  or  more  of  the  given  numbers  hy  any 
prime  romraon  measure;  take  the  qiwfients  and  the  undivid- 
ed numbers  for  a  new  set ;  divide  two  or  marc  of  theAii  by 
(my  prime  common  measure;  and  ^o  on,  uniil  the  resulting 
numbers  are  prims  to  each  other;  the  eontiniied  product  oj 
the  resulting  numbers  and  all  the  divisors  loill  he  the  least 
common  multiple  of  the  gicen  numbers. 

Ex.  2.  Find  tlie  least  common  multiple  of  3C,  120,  and 

216. 

60 


LEAST    COMMON    MULTIPLE.  §107 

36=2. 2. 3 J^  §  l^'^-    Model. — iiesolve    06 

126=2.    3.3.7  '^^^^^    ^^^   prime  factors.    (§99.; 

2IQ 2.2.3.3  2.0  3Sr=2.2.3.3.     Resolve   12G  into 

wwoonoo     if^ia       its  prime  factors.      120=-2.3.3.7. 
^.Z.6.6.i.4.6~l0l^       Resolve  216  into  its  prime    fac- 
tors.    216=2.2.3.3.2.3.-^2.2. 3. 3. 7. 2. 8=^1512.    1512  is  the 
least  common  multiple  of  the  given  numbers. 

Explanation. — The  prime  factors  are  arranged  as  in 
^  104,  and  one  factor  is  taken  from  each  column,  whether 
full  or  not. 

Rule. — jResolve  eacliof  tJic  given  numbeni  into  its  prime 
factors ;  multiply  together  all  the  factors  of  the  largest  nitiTi' 
bcr,  and  all  the  factors  of  the  other  numbers  that  arc  not  fourid 
hi  the  largest  number  ;  the  product  xcill  bn  the  lea^t  common 
multiple  of  the  given  numbers. 

Either  of  the  above  methods  may  be  used  in  the  following 
exercises. 

Ex.  3.  Find  the  least  common   multiple  of  5,  6,  and   7. 

4.  Find    tlie    least   common  multiple   of  2,  4,  6,  8,   12, 

and  16.  L.  C.  M.  48. 

5.  Find  the  least  common  multiple  of  3,  6, 9,  12,  and  18. 

L.  C.  M.  3G. 

6.  Find  the  least  common  multiple  of  5, 10,  12,  and  15. 

7.  Find  the  least  common  multiple  of  6, 12, 24,  and  48. 

L.  C.  M.  48. 
S.  Find  tlie  least  common  multiple  of  8,  24,  and  72. 

L.  C.  M.  72. 
9.  Find  the  least  common  multiple  of  3,  9,  18,  and  72. 

10.  Find  the  least  common  multiple  of  2,  3,  4,5,  6,  10,  12, 
15,  and  20.  L.  C.  M.  60. 

11.  Find  the  least  common  multiple   of  3,  5,  7,  and   11. 

L.  C.  M.  1155. 
61 


>ji07  iT,^TR\CT    NUMBER?. 

l^.  I\..:\  -A.^  iy^vfnurm    multiple  of   2,3,4,6,8.12, 

and  24. 
13.  Find  the  least  common  multiple  of  3, 7,  and  18. 

L.  CM.  '^73. 
11.  Find  tlie  least  com,mon  multiple  of  2,  4,  7,  and   14. 

.  \  L.  C.  M,  28. 
15.  Find  the  least  common  ninltiple  of  3,  5,  15,  and  30. 
10.  Find  the  least  common  multiple  of  2,4,8,  16,  and  32. 

L.  C.  M.  32. 

17.  Find  the  least  common  multiple  of  3,  4,  6,  8,  and  9. 

L.  CM.  72. 

18.  Find. the  least  common  multiple  of  2,  3,  6,  and  9. 

19.  Find  the  least  common    multiple  of  4,  6,  8,  12,  16, 

and  32.  L.  C  M.  96. 

20.  Fi  id  the  least  common  multiple  of  2,  4,  5, 10,  and  '^0. 

L.  C  M.  20. 


PROMISCUOUS  PROBLEMS. 


1.  Read  279301682038040. 

2.  Read  i207§008u40009750. 

3 .  Write  twenty-seven  billions,  three  hundred  and  three 
millions,  four  hundred  and  seventy-five  thousand,  and 
eighty-nine. 

4.  Write  five  hundred  and  five   billions,  and  fifty-five. 

5.  Add  3  millions  24  thousand  and  17,  4  hundred  thou- 
sand 7  hundred  and  98,  4  millions  247  thousand  and  56, 
and  724  thousand  8  hundred  and  29.  Sum,  8396700. 

6.  Add  twenty,  2  hundred  and  2,  2  thousand  and  27, 
20  thousand  278,  202  thousand  7  hundred  and  89,  and  2 
millions  27  thousand  8  hundred  and  90. 

§2 


PROMISCUOUS   PROBLEMS.  §107 

7,  UnB  9  millions  and  9,  subtract  5  iiu]Iiaub789 

adred  and  54.  3216355. 

8.  From  80  milliaiis  85  thousand  and  8,  subtract  65  inil- 
•liops  764  tboueand  3  hundred  and  "^ '  Kem.  143206o'3> 

0    Multiply  4  hundred  and  70  i  '  "^  ^ "'"■••'  •  r.d 

!■  thousand  8  (mndred  and  1 . 

10.  Multiply  90  thousaud  7  hundred  aud'S,  b}^  80  tliou- 
sand  Ghundred  and  4.  Prod.  7::nilS5J^20. 

11.  Divide  2  billions  59  million'   '    '   -  -ousand  and. 72, 
50  thousand  7  hundred  and  9.  Quot,  40608. 

12.  ]'>ividc  8  billions  777  millions  887  thousand  5  hui:- 
dro  31,  by  97  thousand  5  hundred  and  31. 

13.  The  minuend  is  4  hundred  thpusand  4  hundred  ;  the 
subtrahend  is  364  thousand  7  hundred  aud  26  :  what  ks  the 
remainder?  -  '  Ans.  35674. 

14    The  minuend  is  57  thousand  and  57  ;  the  subtrahend 
27  thousand  5  hundred  and  79  :  what  is  the  remainder  1' 

An.'i.  mis. 

15.  The  minuend  ic?  75  thousand  and  6^i ;  the  remainder 
■  •  36  thousand  and  57  :  what  is  the  subtrahend  1 

16.  The  subtrahend  is  3  millions  and  75  ;  the  remainder 
I'j  5  hundred  thousand  7  hundred  and  5  :  what  is  the  minu- 
end ?  Ans.  3500780. 

17-  The  remainder  is  777  thousand  7  hundred  and  7 ; 
t-hc  subtrahend  is  654  thoiis.fnd  3  hundred  and  25;  what 
is  the  minuend  ?  Aus.  1432032. 

18.  The  multiplicand  is  3  millions  and  75  ;  the  multi- 
plier is  5  hundred  thousand  7  hundred  and  5  :  what  is  the 
product  ? 

19  The  multiplier  is  3  thousand  3  hundred  and  3  ;  the 
multiplicand  is  75  thousand  4  hundred  and  25:  what  is  the 
product  ?  Ans.  249128775. 

63 


§107  ABSTllACT    NUMBERS. 


20.  The  product  is  670  millions  592  thousand  745  ;  the 
multiplier  is  12  thousand  845:  what  is  the  multiplicand  T 

Ans.  54321. 

21.  The  multiplicand  is  40  thousand  5  hundred  and  6  : 
•Le  product  is  413  millions  282  thousand  7  hundred  and 
18  :  what  is  the  multiplier  ? 

22.  The  dividend  is  1  billion  546  millions  2(')3  thousand 
5  hundred  and  4;  the  divisor  is  71  thousand  2  hundred  and 
17  :  what  is  the  quotient?  Ans.  21712. 

23.  The  dividend  is  2  billions  162  millions  6  hundred 
thousand  ;  the  remainder  is  19  thousand  4  hundred  and  90  ; 
the  quotient  is  24  thousand  and  6  :  what  is  the  divisor  ? 

Ans.  90085. 

24.  The  divisor  is  14  thousand  and  20  ;  the  quotient  is 
2  thousand  3  hundred  and  45  :  what  is  the  dividend  ? 

25.  The  divisor  is  7  thousand  and  2  ;  the  quotient  is  2 
thousand  and  7  ;  the  remainder  is  2  thousand  and  7  :  what 
is  the  dividend  ?  Ans.  14055021. 

20.  Resolve  3  thousand  and  80  into  its  prime  factors. 

P.  F.  2,2,2,  5,  7,  11. 

27.  Resolve  5  thousand  4  hundred  and  GO  into  its  prime 
factors. 

28.  Resolve  4  thousand  and  4  into  its  prime  factors. 

P.  R  2,2,7,  U,  13. 

29.  Find  the  greatest  common  measure  of  58,  87,  and 
2610.  G.  C.  M.  29. 

30.  Find  the  greatest  common  measure  of  118,  177,  and 
295. 

31.  Find  the  greatest  common  measure  of  4^',  80,  128, 
and  176.  Q.  C.  M.  16. 

32.  Find  the  least  common  multiple  of  3,  7,  9,  12,  and 
18.  L.  C.  M.  252. 

64 


COMMON    FRACTIONS.  §109 

33.  Find  tbo  least  common  multiple  of  2, 5,  8,  11,  and  14. 

84.  Find  the  least  common  multiple  of  2,  4,  7,  11,    16, 
and  22.  i  L.  C.  M.  1232. 

35.  What  number  is  that  to  which  if  1231,  8912,  5G78, 
45G7,  and  9123  be  added,  the  sum  will  be  47275  1 

Ans.  17761. 

31.  What  number  is  that  from  which  if  1234,8912,5678, 
45li7,and  9128  be  subtracted,  tlie  remainder  will  be 47275? 

37.  What  number  is  that  by  which  if  9876  be  multiplied, 
the  product  will  be  121919220?  Ans.  12345. 

38.  What  number  is  that  by  which  if  5483886  be  divided, 
the  quotient  will  be  2468  ?  Ans.  2222. 


FRACTIONS. 


§  108.  A  FRACTTON  is  a  part  of  a  unit.  Thus,  one  half, 
three  fourtlis,  two  fifths,  five  sixths,  four  sevenths,  three 
eighths,  five  ninths,  and  seven  tenth-i,  are  fractions. 

Fractions  are  of  two  kinds,  Common  and  Decimal. 


COMMON  FRACTIONS. 

§  109.  A  common  fraction,  or,  simply,  a  fraction,  is  de- 
noted by  two  terms!,  one  above,  and  the  other  below,  a  hori- 
zontal line.  The  term  above  is  called  the  numerator  ^  the 
term  below  is  called  the  denominator.  Thus,  the  above 
fractions  are  denoted,  i,  f ,  f ,  |^,  f,  f,  -|,  ■^.  The  numera- 
tors are  1,  3,  2,  5,  4,  3,  5,  and  7  :  the  denominators  are  2, 

4,  5,  6,  7,  8,  9,  and  10. 

E  65 


§110  ABSTRACT   NUMBERS. 

Point  out  the  numerator  and  tlie  denoujinator  of  eacli  of 
the  following  fractions  :  i,  -|,  f,  ^,  f,  |,  |,  -V»  -rr»  T2»  tt> 

5  7_     _P_         3        _R_     _2_     _3_     _4_     _J>_     _fi_      _7_     _8_     _n_     lH     1  I      1^ 

14"     15»    10'    17'    18'     19'    20'    2  1'    2  2'    23'    2  4'    2  3>    26»    37'26>3»> 

iA     i±     ^5-    JL1      <^      JLS       a  o  0      3_7 4-  "^       2^JLF 

3^'     31'    32'    3 '3'    3  4'    35'     1200'    2000'    376'    9  72* 

§  110.  A  fraction  is  rf-ad  by  |>ruinnniciiig  after  the  nu- 
merator the  ordinal  of  the  denominator  iti  the  singular  or 
the  plural  number  accorditig  as  the  numerator  is  «»ne  or 
mora  than  one.  Thus,  -t  is  read,  one  fifth  ;  |,  two  fit's  hs; 
^,  three  twenty -firsts  ;  -g^,  four  thirty-.-ecoMds  ;  ^^^17,  five 
two  hundred  and-ninths;  -^jy^^,  t^ix  two  thoui^and-a nd- sec- 
onds ;  -g^y'og^,  seven  three  thoiisand-one-hundred-and  .-ixths. 

But,  if  the  denominator  is  2,  the  fraction  is  read  half  or 
halves,  and  not  second  or  seconds.  Thus,  -},  one  half;  -f, 
thrre  halves. 

IIj^'kI  thp  followinfy  •   ^   ^-    *    —    "     -'     "_   JL.   _+_  _p_  _i_     s  ,_ 

4  ^-        _6_      __?_      _8_      _9_       1  O.      _:i_      _4_      _S_      _6_      _7_      _8_      _?i_     J  D      1   1 

IT'     16'    17'     18'     19>    20'    21'    22'    2  3'    24»    25'    2  6'    2  7»    28»2  9'30> 

12       1  3      X±     XJi     -L<>      11.     -lA     JL^l 
ST'    32'    33>    34'    35'    30'    37'    3b' 

§  111.  A  fraction  \^  prod  me  d  by  dividing  a  unit  or  a 
iQunjber  into  some  nuniber  of  e<|ual  f»arts.  Thus,  \  is  pro- 
duced by  dividing  the  unit  into  four  etjual  parts:  f,  bj 
dividing  2  into  5  equal  parts  ;  ^,  by  dividing  5  into  9  e(jual 
parts. 

The  nun)erator  is  the  divi<lend,  the  denominator  is  the 
divisor,  and  the  value  of  the  fraction  is  the  quotient.  See 
$48. 

How  is  I  produced  ?   |-?  |?  i?  -,%  ?  ^?  A?  i^?  ^? 

VP  ?    iJ.  ?    u  ? 

Xs •       2  4'       35* 

§  112.  Otherwise,  a  fraction  may  he  produced  by  divid- 
ing a  unit  into  some  number  of  equal  parts  and  con.-tder* 
ing  either  one  or  several  of  thesb  parts.  Each  of  these 
parts  is  called  a  ^//ar/ />;/<«/ 7/1* /V  ;  and  a  fraction  is  f^irher 
One  or  several  fractional  units.     The  denominator  shows 

C6 


COMMON    FRACTIONS.  ^113 


into  liow  many  partes  the  unit  is  divided,  and  the  imnierator 

fihows  iiow  luany  parts  there  arc  iti  the  fraclii)ii.     Thus,  iit 

|,  one  fifth  is  the  fractiotial  unit,  and  the  fraction  containg 

three  of  these  units  ;  in  ^,  the  fractional  unit  is  one  niutli^ 

and  the  fraction  contains  seven  of  them. 

In  this  view,  how  is  ^^^  produced  ?    3^^?  {r  ?  f?  J^  ?   y\f 
7    ?   _S    ?   j<  ?    2  ?   _5_  ?    _«_? 

« 4   •      a  r>  •      7   •      9   •      11"      10* 

§  113.  The  o  t/iifi  of  a  fraction  is  the  quotient  of  its  nu- 
merator divided  by  its  denominator.  Tliis  value  depends 
on  the  value  of  tjie  fractional  units?,  as  well  as  <u\  the  niun- 
her  of  tliein.  If  the  fractional  units  of  sevt-ral  fraeti.mt 
are  ef|ii:jl,  of  course  the  greatest  fraction  is  the  one  which 
has  the  most  fractional  units.  Tiiat  is,  if  the  denoui'nutors 
arc  e(|nal,  the  groate.^'t  fraction  is  the  one  wisieli  has  tho 
greatest  numerator.  Aga'n,  if  the  nuniher  of  frictional 
units  in  several  fraction!*  is  the  same,  of  ct)urse  thegieiresl^ 
fraction  is  the  one  which  has  the  largest  fraotiMia!  nu'ib^ 
But,  tiie  larger  the  number  of  parts  into  which  a  unit  i^ 
dividt^d,  the  sn>aller  each  part  must  be.  Therefore,  if  tho 
uumiM-ators  of  ^several  fractions  are  equal,  the  greatest  frac- 
tion is  tlie  one  which  has  the  smallest  den(Mninator. 

How  do  I  and  *  compare  in  value  ?    -^-  and  f  ?  y  and  ^t 

r-  and  ^  ?  ^V  »»<1  t*t  ?  A  a"^  t\  ?  xV  »" ^  t\  ^  U  ^'»"J  i^«  ^ 
^  and  I-  ?  -/3  and  -,\1  -,%  and  A?  t  and  ^  ?  -*  and  /j-? 
\i-  a"<l  M?  U  «nd  H?  li  and  f^?  ||  and  fi?  ^and^f 
From  the  definition  in  §  1U8,  the  number  of  fractional 
units  in  a  frae;ion  must  be  less  than  the  number  of  pjirti 
into  which  tlie  unit  is  divided  ;  that  is,  the  numerator  must 
be  less  than  the  denominator.  Larger  numbers,  however, 
may  be  expressed  in  a  fractional  form  ;  and  such  expres- 
siocs  are  improperly  called  fractions  also.  Ilejce  the  fol- 
lowing distinctions: — 

67 


§114  ABSTRACT   NtJMBERS. 


§114.  A  proper  \rnct\im  h  owe  whose  numerator  is  Icsm 
than  its  <lon»»niinator,  antl  wh/»se  value  is  less  ihnn  a  uirit. 
Tl'iis  I,  ly  ^,  -♦,  J.,  I,  A,  A,  _z^,  _«_^  .7^.^  and  |.},  arc  proper 
fractious. 

§  Ho.  An  improper  fraction  is  one  wlioFe  nunicrator  if 
not  /r.v.s-  tlun  ita  (Iciioniiniitor,  jind   whose  value   is   not  ]e$n 
than  a  unit.     Thus,  -?,  t,  2,  {-,  §,  ^  V,  V,  Vj  Vu',^*"^  -f^ 
^«ir«  jinpr(»per  fractious. 

:I.s  I  a  proper  or  an  improper  fraction  ?     *  ?   ^  ?  -|  ?    -^\T 

JL;  ?        nV      11^      12?      17?      209      20   9      W?        ft     ?      P?     »V      f»?      7^ 
*  1  ■      J,;J  •       ♦    '       &    •     'JO'      1  >   *     2  5  '      1  5  •     1  o  •     w  •     ¥  •     y  •     w  * 

§  lJ(j,  A  unit  is  often  ouilcd,  for  distinction,  un  iTif/yral 
IBnit  ;  and  a  number  of  integral  units  is  called  an  integer, 
.er  an  inlegral  number. 

§  117.  A  mixed  number  is  one  composed  of  an  integer 
and  a  fraction.  Thus,  5^,  read  five  and  one  half,  is  a 
mixed  number.     So  al.-o,  (i^,  3^,  18^,  31;,  GG^. 

A  fraciional  unit  or  a  fraction  may  be  divided  into  anj 
nuuilMjr  of  equal  parts.  Thu'^,  if  I  be  divided  into  o  equal 
part.",  Ciich  of  the  parts  is  J  of  ^.  So,  if  ^  of  I  be  divided 
into  4  equal  parts,  3  of  these  parts  arc  ;^  of  ^  of  ^.  Such 
cxpiev-ioiJB  are  called  comjiouiid  fractions.     Jlencf, 

§  118.  A  compound  fraction  is  a  fraction  of  a  fraetioiL 
Thu.,  I  of  -^  i  of  vV,  I  of  },  l],  of  12X,  and  ^  of  ^  of  33^ 
are  a)njj)ound  fractions., 

§  119.  A  complex  fraction  is  one  which  has  a  fraction  for 

2 
its  numerator  ojr  its  denomitator  or  each.     Thus,  -    ,    read 

two  divided  by  five  and  one   half,  is   a   complex  fraction, 

6i   12i-    .}  of  I  \  of  18J 

*^"  "^'''  y '  37i*  "33i  '  vTf  12.i- 

68 


DEDUCTION   OF   COMMON    FBACriONR.  ^123 


re:  UCTION  OF  COMMON  FRACTIONS. 


§  120.  T()  reduce  any  mimbcr.  cither  fracti)nar  or  inte- 
gral, i^  to  cha'ige  its  /'ftni  of  expression  wlthmt  r/i>n>f/ing{ 
ifs  Vdfiir,  Thus,  a  unit  ni>i)'  be  reduced  to  *,  or  to  y,  or  tfll 
I'o"  • — i  '"^^^  ^'^  reduced  to  ®,  or  to  -}-i,  or  to  ^\  : — (j\  nja v  be 
reduced  to  ^^,  or  to  ^£  : — and  y  ,iiay  be   reduced   lo  12^. 

§  121.  A  fraction  is  in  its  lowest  terms  \vht;n  its  tcrrafl 
are  pnuie  to  each  other.  (^  04).  Thus,  -^,  \^  |,  -y,  ;^,  -J, 
and  'J',  are  each  in  its  lovve-t  terms. 

Ex.  1.  Reduce  \^  to  its  Newest  term?. 

4^;if^l4:i^'9j^|i  ^  122.    MuDKL.— Divide   botk 

terms  by  4  :  4  in  14  t,  3()  times; 
4  in  .^TG,  144  times:  divide  both  these  quofit-nts  by  4:  4 
in  oG,  !>  times;  4  in  144,  oG  rime-;:  divide  both  fho.»o  <juo- 
tienrs  by  9  ;  9  in  9,  once  ;  9  in  HG,  4  times.  'J'h<^vo  (|U0- 
tients  are  prime  to  each  other  :  hence,  \  is  the  given  Irao- 
tion  in  its  lowest  terms, 

]Cxri,AVAT!<>N,— By  eompHririg§^  77  and  111,  it  is  evi- 
dent th  it  the  value  of  a  Iracfioii  i>  not  changed  »iy  dividing, 
both  its  ternjs  bythesame  nnnjber:  and  by  jsucci^s.sive  divi- 
sions, its  terms  may  alwa\s  be  made   piiujc   to  each  oiher. 

Ex.  -.    lied  nee  -^^' !-  to  its  lowest  term^. 

2v  -llf^ll^  §  128.   M(n.Ki,.  — Find  the  greatest  com- 

mon measnre  of  the  terms  of  tln^  fiacfion. 
(§  108i.  Their  greatest  con)nn'ri  measuie  is  27.  D.vide. 
both  terms  bv  27  :  27  in  5G7,  21  times;  27  in  67;'),  25; 
times.  21  twenty  iitths  is  the  given  fraciion  iii  its  iowest 
terms. 

Rui-R  for  reducing  a  fraction  to  its  lowest  terms. 

1.  Dciilii  hotk  tf^.rm<  hj  (till/  conitiiim  mnsan;  dioidt 
Loth  t/iKSf  (jU'ffifnfs  })ij  ttiii/  cninniou  inensnrr  •  (dh/  .-o  itti^ 
Ultlll  ihi'  qii'itif.uf>i  arc  in'Im.f  to  r<irh  nlhr  ;  (lit:  lual  (juuhcHfj 
will  tfS  the  l(jioeil  term.<  of  the  ij'ocn  fractiun. 


|124  ABSTRACT    NUMBERS. 


Or,  2.   Divide  lofh  ferrns  It/  tlidr  f/rrafcst  cornnion  iiiea*- 
urf  :   the  qnofiniis  will  he  the  loua^t  lams, 
Ex.  3.  Reduce  -^l^^^  to  its  lowest  termis. 

4.  Reduce  f  i^^  to  its  lowest  terms.  Value,  •^. 

5.  Reduce  ^|-^  to  its  lowest  terms.  A'^al.  |-, 
G.   Reduce  -*-f^-  to  its  lowest  terms. 

7.  Reduce  |f^-2-  to  its  lowest  terms.  A'nl.  -jV 

J^.  Reduce  \-^^\  to  its  lowest  terms.  Yal.  \\^ 

P.  Reduce  -^—-r  to  its  lowest  terms. 

18  9  4 

10.  Reduce  f-|{)^  to  its  lowest  terras.  Yal.  ^, 

11.  Reduce  -|^|-f  to  its  lowest  terms.  Yal.  -/-j^ 

12.  Heduce  ^-^-^-l  to  its  lowest  terms. 

13.  Reduce  -ff-^  to  its  lowest  terms.  Yal.  l^^, 

14.  Reduce  ||-^  to  its  lowest  terms.  Yal.  -^^^ 

15.  Reduce  ^44^4-^  to  its  lowest  terms. 

10.   Hedtice  ^ffl  to  its  lowe-t  terms.  Yal.  ^^^ 

17.  Reduce  -^Vs  ^^  '^^^  lowest  terms.  Yal.  t/qV 

18.  R<duce  14x1  ^^  ^^^  lowest  terms. 

19.  Reduce  |f?-i  to  its  lowest  teims,  Yal.  f?-^^ 

20.  Reduce  -^-«ff^  to  its  lowest  terms.  Yal.  |^-^ 

21.  Tn  Ytt*  ^f'W  mapy  units  i' 

8G8'16  §  124.  MoDKL.— Divide  tlie  numer- 

f58  ,54tt=«'>4-}        at(»r  })y  the  deiiomiuafor.  (?^  71).    Thb 

4,  rpiotient  is  f)-4,  and    tlie   rtniaindor   4, 

Redufo  -.-V  to  its  luwest  teruiS.     The 

jiren  fraction  is  equal  to  54^. 

KxriiANATiON. — Since  IG  sixteenths  make  a  unit,  the 
BTimher  of  units  in  SG*"^  sixteenths  is  ecjual  to  the  number 
pf  limes  IG  is  contained  in  8C8.     See  §  U.S. 

Ri'LK  for  reducing  an  improper  fraction  to  a  whole  or  a 

nixed  number. 

/ 

70 


DEDUCTION   OF   COMMON    FRACTIONS.  §125 


D'lvlde  the  numerator  bij  the  ilenominafffr  ;  the  qKotient 
will  he  the  integral  part.  Place  the  denondnator  umUr  the 
rem ai inter  for  the  fracti  mul  part, 

Ex.  22.    [n  y*  how  ininy  units?  Anf*.  25. 

23.   Ill  Y  l^ow  many  units?  Ans.  24.1:. 

2\.   In  -i--^—  ^'<J^^  many  units  ? 

25.   In  y  liow  many  units?  Ans.  3-2-. 

28.  Reduce  J-«n.  to  units.  Val.  20. 

27.  Reduce  ^-^.s>  to  units. 

28.  Reduce  \-Y  ^^  units.  Val.  14-*-. 

29.  Reduce  V/  t^>  ufiits.  Val.  8^. 
80.  Reduce  ^.-y  to  units. 

31.  Reduce  W^  to  a  njixed  number.  Val.  134. 

32.  Reduce  Y/  ^0  a  mixed  number.  Val.  12^^. 

30.  Reduce  "Yt^  ^0  a  mixed  number. 

84.  Reduce  y^^  to  a  mixed  number.  Val.  11-^V 
S5.  Reduce  y^  to  a  mixed  number.  Val.  10-^-. 

36.  Reduce  ^-^^  to  a  mixed  number. 

37.  Reduce  y/  to  a  mixed  number.  Val.  S^^. 

85.  Reduce  ^^^  to  a  mixed  number.  Val.  *7~\ 
S9.  Reduce  \\/  to  a  mixed  number. 

40.  Reduce  VV  ^^  a  mixed  number.  Val.  69V 

41.  Reduce  10  units  to  fourths-. 

§  125.   MiM)Kf..— Multiply  4  fourths  by  10. 
10=*jp     The  product  is  40  fourths  :  hence  10  uLits— 
40  fourths. 

■  ExPLAN.VTioN.— Since  4  fourths  make  a  unit,  10  units= 
10  times  4  fourths;  that  is,  40  fourths. 

Rui.K  for  reducing  a  whole  number  to  any  fractional  de- 
nominiition. 

Mif/tlpij/  the  nu'^ber  of  fractional  units  in  a  unit  by  the 
number  of  units. 

71 


§126  ABSTRACT    NUMBERS. 

Ex.  42.  In  3  units  how  many  fifths  ? 

43.  In  5  units  how  many  sevenths?  An?.  ^^. 

44.  In  6  units  how  many  ninth;^  ?  Ans.  '^, 

45.  In  7  units  how  many  elevenths? 

46.  Ueduce  8  to  thirteenths.  Val.  VV- 

47.  Reduce  9  to  fifteenths.  A'al.  VV. 

48.  Reduce  11  to  seventeenths. 

49.  Reduce  12  to  twentieths.  Val.  ^*s, 

50.  Reduce  13  to  twent-   ^  nrths.  Val.  W. 

51.  Reduce  14  to  tweni^-i.iDths. 

52.  Reduce  15  to  thirty-fifths.  Yal.  •'W' 

53.  Reduce  10  to  a  fraction  with  denominator  40. 

Val.  Vo^ 

54.  Reduce  17  to  a  fraction  with  denominator  4H. 

55.  Reduce  18  to  a  fraction  with  denominator  53. 

Val    ^-^ 

56.  Reduce  i9  to  a  fraction  with  denominator  59. 

Val.  ^}-l^, 

3  0 

57.  Reduce  20  to  a  fraction  with  denominator  05. 

58.  Reduce  21  to  a  fraction  with  denominator  71. 

Val   ±±SJL, 

'  59.  Reduce  22  to  a  fraction  with  denominator  77. 
60.  Reduce  23  to  a  fraction  with  detiomiuator  85. 


61.  Reduce  16^-  to  an  improper  fraction, 

16i  =  V  +  2  =  V         §  ^-^-  MoDKL.— Reduce  16  units  to 
halves.      Add  o2  halves  and  1    half. — 
The  sum  is  33, halves:   hence  16^  is  equal  to  33  halves. 

Rule  for  reducing  a  mixed  number  to  an  impro[>er  frac- 
tion. 

Rtduce  the  infeger  to  the  (JcnomI nation  of  the  fraction  ^ 
add  the  two  numerators  tojefher,  and  under  their  sum  mttthe 
common  denominator. 

72 


REDUCTION   OF   COM  MOM    FRACTIONS.  §127 


Ex.  Ci.  Reduce  3|  to  tliiids.  Val.  y. 

G-J.   R«^.(iuce  4.^-  to  fourths. 

64.  Re<luce  61  to  fifrlis.  V;il.  \K 

65.  Ri-duce,  8j  to  sixihv.  •  Val.  *^*. 

66.  Rod  wee  lOy  to  sevenths. 

67.  Ke(]uoe  12.^  to  eighths.  Val.  JlOJ.. 

68.  Reduce  14.}  to  ninths.  Yal.  -L^-i 
60.   Reduce  16 tV  to  tenth.*. 

70.  In  17^1  ^i"^v  many  elevenths?  An^.  yy. 

71.  In  ISfV  hovf  many  twelfths? 
7-j.   In  19yV  '^*^^  many  fourteenths? 
7'1.   In  2)i\  how  many  sixreenths? 
74-.   In  21/^  how  many  eighteenths? 
75.   In  22,/ ^-  how  many  tA^eiity  firsts? 
70.   Reduce  23^:^-  to  an  improper  fraction. 
77.   Reduce  24.}"-  to  an  im[>roj)er  fraction. 
7H.   lleduce  25/^  to  an  improper  fraction, 
79.   Rfdiice  523^3  to  an  improper  fraction. 
8u'.  Reduce  65 y^-  to  an  improper  fraction. 

81.  R-jduce  -•'-  to  twentieths. 

•J! 

^=y^;         ^1-7.   Model. — 5  twentieths  make  one  fourth, 
^lultiply  both  terms  hy  5:   5  times  3  arc    15;  5 
times  4  are  20.     a  js  equal  to  15  twentieths. 

Exp.tiANATioN. — By  comparing  §.!^  76  and  111,  it  is  evi- 
dent that  the  value  of  a  fraction  is  not  changed  hy  multi- 
plying both  its  terms  by  the  same  number.  We  divide  the 
re(]uired  denominator  by  the  given  one,  and  multiply  both 
terms  of  the  fraction  by  the  (juotient. 

RiiLK  for  reducing  a  fraction  (o  a  larger  denominator. 

AInitipf(j  infh  ferma   hj   (lie   qwitient  of  the   rKjuiitd  dc' 

nomijKtfor  divided  bj/  fhe  ijlccn  one. 

73 


Ans. 

2  2  3 
"1  u  • 

Ana. 

3  2  5 
1  6   * 

Ans. 

3  8  S 
16  * 

Yal. 

R  n  n 
a  *  ♦ 

Yal. 

6  7  8 
U  7    * 

Yal.  1 

718 
3  3     • 

Yal.  4 

"281 
6  »  '  • 

§128  ABSTRACT    NUMBERS. 


Ex.  ^2.  Hednce  |  to  tenths.  Val.  yV 

80.  Heduce  A  to  eiglitecntlis.  Yal.  i-5,^ 

84.  Eednce  -f  to  tbirty-fifilis. 

85.  R.duce  |- to  forrieths,  Val.  J^|^ 
8(>.  T^educo  |-  to  sixty-thirds.  Val.  -|-|v 

87.  Reduce  —o  ^^  ninetieths. 

88.  R«*diice  ^  to  ninety-ninths.  Val.  ^. 
81).  Heduce  ^^  to  sixtieths.  Val.  f^. 
90.  Reduce  -^3  to  sixty-fifths. 

9i.  Reduce  yoi;  to  twentieths. 

§  128.  Model. — Divide  both   terms  by  5c 
5]_^.'^_j_^     5  in  2o,  5  limes  ;  5  in  100,  20  timccs.     25  one 

bundredths=:5  twentieths. 
For  explanation,  see  §  122. 

Rule  for  reducing  a  fraction  to  a  lower  denominator. 
Divi'le  both  terms  by  the  (jiiotlent  of  the  given  denoinina- 
tor  dii't  led  btj  the  required  one. 

Ex.  92.  Reduce -/zr  to  fourteenths.      x  Val.  ,^-. 

9o.   Reduce  -i§  to  tifteenths. 

94.  Reduce  f^- to  sixteenths.  Val. -/g~> 

95.  Reduce  -^^-^  to  eighteenths.  Val.  -^^. 
9().   In  -^^-^  how  many  twentieths? 

97.  Ill  f-^-  how  many  twenty  fiists  ?  Ans.  ^^w 

98.  In  -ifi  how  many  twenty-thirds?  Ans.  ^^. 

99.  In  -~V  ^^^^  many  twenty-fourths? 

100.  In  22o-  ^^'^^v  "^^^"y  thirtieths?  Ans.  -?-§.. 

101.  Reduce  -,},  ^:,  and  |,  to  a  common  denominator.  (§91). 
^  ,  ,  §  129.  iMoDKL.  —  Multiply  both  termn 
''^        *        ''^      of  the  liist  fiuciion  by  Ji2 ;    32   times  1 

•ff     ^%     oi     ^'"^  '^-;    ^-  times  2  are  04:     multiply 

both  terms  of  th»'.  second  fraction  h)  16; 

16  times  0  are  48  ;   ]G  times  4  are  64  :  multiply  both  termi 

74 


PPBUCTION    OF   COMMON    FRACTIONS.  §129 


of  tlio  Oiird  fr.-iotion  b}'  8  ;  8  times  5  nre  40  ;  8  tiiues  8  are 
64.  The  j?ivpn  fractions  are  respectively  cijual  to  3J,  48, 
and  40  sixty- fuurtb.". 

ExrrANATiON. — The  values  of  the  fractions  are  not 
chart|j(H],  because  both  terms  of  each  fraction  are  multiplied 
by  the  f-anie  iiuiuber:  and  the  denominators  are  alike,  be- 
cause each  one  is  produced  by  mulfiplyinc  together  all  the 
given  denojiiinators.  The  n)ulti[)lier  82  fur  the  first  frac- 
tion is  4  X  8,  the  product  of  the  other  two  denominators. 
And  so  for  the  other?. 

RiJLK  tor  reducing  fractions  to  a  common   denominator. 

Mnlnplij  both  ferms  of  euch  f taction  It/  the  product  of  th* 
other  dinnminotors. 

Ex.  102.  Heduce  -|,  -^,  and  -^,  to  a  common  denojiiinator. 

103.  lleduce  -^,  -^-,  and  J,  to  a  common  deooniinator. 

Vi*  I       1  "^       10        8 
*  *"•    3  0'   To'    311- 

104.  lleduce  J,  2,  and  |,  to  a  common  denominator. 

V-i  I        2  0       4  5       a  4 

105.  Ilt>duee  -^,  |,  and  -^,  to  a  common  denominaror. 
100.   Reduce  *,  ^,  and  |-,  to  a  common  detioniinator. 

'  "' •   2  10'   a  1  0 »  a  1 6^ 

107.  Reduce  -\^  f,  and  y,  to  a  common  denominator. 

Vi  1     -'■  «       _f  fi_     JL?J» 
'   '    '•     3  3  (>'     a  3  6'    330* 

lOH.  Re<lnce  i,  ^,  and  -y,  to  a  common  denoniibutor. 

109.  Reiluce  ^5  I,  and  -j'\j,  to  a  commoti  denominator. 

V-il        ^(^        1  «<'      2JL0 

*  «J ' .  -7  [.  o  1  Tii  0 "  ^7  a  0* 

110.  Reduce  -^,  -^^,  -^\^  and  gV,  to  a  common   deiionjina* 

tof  V'il     _2>oo.      _8 !_« Q_    _finoo_       7  s  6  O ^ 

'"•2  16do'»216OO'2  1C0O*21G0O" 

11  '.  Ri'iluee  I,  -j^j,  and  -j^g-,  to  a  common  dtMjominator. 
111.'.   Roduee  -),  1^,  and  y'j-,  to  a  common  denominator. 

'   "    •    «  1>  3  »    G  J  3  '    a  9  3  * 

lis.  Reduce  -j^,  ^,  and  -fo-».^^  ^  common  deitoniinaioi-. 
114.  Reduce  ^,  fj  '^^d  *.  to  a  cunimon  deuumiiiittor. 

75 


§130  ABSTRACT    NUMBERS. 


115.  Ileduee  \,  ~,  and  ^-,  to  a  common  dononiinator. 

Vj,  I         5  t  3  6  4.» 

*  '*  •  "?!¥'   2T6>  TT9' 

116.  lieduce  ^,  |,  and  y,  to  a  common  denoiniiuiror. 

V-il     -iL'i-    j^3^    iOO 

*  '"  •    14  0'    1  4  0  »    1  ¥!?• 

117.  lieduee  ^,  |-,  and  |-,  to  a  common  deuominator. 

118.  Reduce  i,  f,  and  f,  to  a  common  denomiitator. 

V-il     -3Jl^     ^7J!        _40 
'"'•   ibO>   ISO'    1  sU* 


119.  Reduce  ^-,  -f-,  and  — ,  to  a  commovi  dennmiii  i  or. 

Vi  1        7  0  SO         1 

*  '"'  "2  8  0'  ^so*    a 

120.  Reduce  ^,  f,  and  y\,  to  a  common  deuominator. 


L       A       A 

2  4  8 


ILM.  Reduce  |,  -|,  and  |^,  to  their  least  common  denomi- 
nator. 

§130.  MoDFL. — Find  tliejenst  common 
multiple  of  the  denominators.  (§  1(J(>).  8  is 
-|  I  1^  their  leist  Ci>ujmon  njulti|tle.  MuUiplj 
both  terms  of  the  first  friicrion  by  4  :  4 
times  1  are  4;  4  times  2  are  8  :  multiply  Ixirh  terni>i  of  the  • 
seco'id  frdc'i  )n  by  2:  twice  3  are  6;  twice  4  are  8;  th« 
third  fraction  is  already  of  tlie  required  dcuoiiiiiiurion. — 
Tlie  given  fractions  are  respectiveiy  equal  to  4,  6,  and  5 
eighrh.^'. 

Exp  t.  A  NATION. — To  find  the  proper  multiplier  for  the 
terms  of  either  f^yiction,  we  divide  the  least  conimon  mul- 
tiple by  its  denominator.     See  §  127, 

RuLK  for  reducing  fractious  to  their  least  common  de- 
noniinafor. 

Fmd  /he  hast  common  multiple  of  the  denominalorii,  and 

rtducc  each  fraction  to  the  diiiomlii'ilioii   expressed  hij  thU 

mulfipf''.      E'ich  fraction  iniist  first  he.  iiL  its  hnct^t  terms, 

Ex.  122.  Pteduce  f,  -4,, and  |,   to  their  least  con  m  m  de- 

\  n  o  I  n  i )  i  a  t  o  r .  Ya  1 .  y\  »  fV?  -]rh 

123.  Reduce  I,  |,  and  ^,   to  their  least  commou  duuomi- 

iiatur. 

7G 


\ 


1?ED€CT10N   OF   COMMON  FRACTIONS  §180 

124.  Il(3duce  J,  -f-,  and  |,  to  tlieir  least  common  dcnoiiii- 
Udior.  vai.  ies,  ins^  lo.s* 

125.  lleduce  ^,  I,  and  y^,  to  their  least  comnKHi  dcMiomi- 
nator.  Val.  -,V  ^\„  H^. 

126.  lleduce  ^.,  ^,  and  -^,  to  tlieir  least  comniuii  dciiomi- 
h,  tor. 

127.  lleduce  -^,  |,  and  ^"^,^5  to  their  least  common  denomi- 
nator. Va!.  -/-, /^,  I-*-. 

128.  Reduce  .},  ^,  ^,  and  i,  to  their  least  common  de- 
nuujiiiator.  Yal.  -j\-,  ^\)  ■^•\,  -^-^, 

129.  Reduce  -^,  i,  ^,  ^,  and  -j\,  to  their  Uaat  common  de- 
nominator. 

180.  Reduce  |,  I,  |,  -|,  and  |;^,  to  their  least  common  de- 
iioiuinator.  ^       Yal.  aa,  -^I,  f^,  -|^.>  |->. 

181.  Reduce  IJ,  2,  >ij  To>  ^'^'^  "sfj  *^  their  least  common 
deiu.minator.  "        Val.  if,  f*-,  |f,  -^,  if, 

132.  Reduce  l,  ^,  J-^,  and  H,  to  their  least  common  de- 
nominator. 

138.  Reduce  -^,  -^\,  ^Vj  ^^^^  2^5  to  their  least  common  de* 
nominator.  Val.  |, -*,  |,  1- 

184.  Rtuliice  I,  I,  -j^^,  and  -\l,  to  their  least  cumuxm  de- 
nominator. Val.  -j'V,  -ft-,  -i«3^,  -j^a^. 

135.  Reduce  -^\,  -^\-,  -^-^^  and  -^\,  to  their  lea.st  common 
deiiomiuator. 

181).  Reduce  I,  -|,  |-,  f,  and  ^,  to  their  least  common  de- 
i^^i"i"ator.  Val.  |§,  >^, -*^, -*C,-6-C-. 

187.  Reduce  -J,  f,  /^,  ^f,  and  |-^,  to  their  least  common 
denominator.  Val.  -J-^,  1%  -i*,  |-o,  if. 

138.  Reduce  i,  a,  *,  |,  3^,  ^'*^,  and  l^,  to  their  least  com- 
mon denominator. 

*139.  Reduce  i,  f,  -*,  y^^^,  and  i^,  to  their  least  common 
dt^nominator.  Val.  H-,  -||,  |f,  -*i,  H* 

14U.  Reduce  J,  «,  -^y  /g,  and  5%,  to  their  least  common 
denominator.  Val.  |^,  -1%,  -^,  /„>  »V- 

77 


§131  ABSTRACT    NUMBERS. 


ADDITION  OF  COMMON  FilziCTIONS. 


Ex.  I.  Add^-,  A,|.,  and|. 

§  181.  Model. — 1  and  3  are 
¥  +  ¥  +  ¥  +  {  =  ¥  =  2      4,JH.d  5  are  9,  ai.-l  7  are  K).— 

lo  eii^htlis  is  eijual   to   2.     The 
Bum  18  2. 

ExPLAMAT'loN. — Since  all  the  fraction-!  have  the  samo 
fractional  uni^,  their  ntKuerators  are  ad  led  \'oi'  the  tiunier- 
ttor  of  tlie  sum,  and  the  comiuoii  deuominator  is  taken  as 
its  defioiuiiiator. 

Ex.  2.  Add  I,  I,  and  -|-. 

3  §  132.  MoDFL. —  Reduce  the  given 

^      *      '^  fractioLs  to  'heir   least  c  Mintioa  de- 

♦  ^fi-|_L=ij~2*      nominator.   (§  l:-^0).     4and<)arel0, 

arid  7  a?e  17.     17  eighths  i.s  e([aal  to 

2^,.     The  f^um  is  2^.. 

ExPi.AVATroN. — It  is  evidently  impossible  to  add  the 
given  fr-ictions  wiihout  reduction.  3  fourths  and  7  eighths 
make  Tieither  10  fourth's  nor  10  eighths.  It  is  not  essential 
to  reduce  to  tlie  Irasf  common  denominator;  but  this  gen- 
erally re<|nires  less  labor  than  to  reduce  simply  to  a  com- 
mon denominator. 

Ex.  3.  Add  241:.  351|,  179|-,  and  187. 

24.V        .l:+i-i-|-  §133.  MooKL.— Re- 

'^  2  I  4  .  i_  ij(_iji         duce   the   fracti.ins  to 

179^         "b'ii"t*tt       H         H  their  least  comm.tn  de- 

1q^  nominator.    (§  130). — 

742 f  2  and  4  are  (5,  and  7 

are  13.  13  eighths  is 
equal  to  1  and  5  eighths,  set  down  ^;  1  and  7  are  H.  and  9 
are  17,  and  I  are  18,  and  4  are  22,  -^et  down  2  :  2  and  8  are 
10,  ant)  7  are  17,  and  5  are  22,  and  2  are  24,  set  down  4; 
t  and  1  are  3,  and  1  are  4,  and  3  are  7.     The  sum  is  742f, 

78 


ADDITION    OF    C  'M3I0N    FRACTIONS.  §133 


Ruri*^, — Reduce  the  fractinns  to  their  least  cotnnto.i  de- 
nomlnutor :  a  Id  the  numerators,  and  under  their  .smn  set 
the  coni'noa  deiomlnttor.  Reduce  the  result  to  iis  lowest 
terms  or  to  a  mixed  number ^  as  the  case  mat/  be. 

Ex.  1.    Ada  I,  I,  and  f.  Sum,  1^-, 

5.  Add  1,1,  A,  and  ±  Sum,  2. 

G.  Add  i,f.A,/;,a„d^. 

7.  Add  f,  f,  f,  and  f.  Sum,  2|-. 

8.  Add-^,  I,  A,  and  |.  Sum,  1a 

9.  Add  1-,  A,  A,  ^,  and  ^ 

10.  Add  -L,  f,  A,  and  >-.  Sun.,2j-5. 

11.  Add^,  ^,  A,  and  A  Sliu.,  2^^^. 
*■-•    ^^'J'l  a»  4>  5>  To>  anu  -j^j. 

l;i.   Add  i   4-,  A,  and  yV  Suiii,  l2-|. 

14.  Add  i,  i.  A,  3Z_,  and  -i|.  Stun,  23V 

15.  Add  A,  ^,  8,  ,A,  a„j  i,. 

16.  Add  A,  ^,  I,  X,  |i.,  and  a^.  S.hu,  4}|, 

17.  Add  A,  I,  3,  -^-,  ^^_.,  and  3^^.  Smn,  2ai. 

18.  Add  A,  I-,  A,  and  «. 

19.  Add  A.  i,  L,  and  -Jj-.  Sun.,  ^«-V^. 

20.  Add  2^,  3f,  4A,  and  5*.  Sim,  16aa. 

21.  Add  4,  8A,  9^,  and  llf. 

22.  Find  ihe  suiu  of  IQa,  21a.  32^-,  43^*^-,  and  54/^. 

Sum,  162^*0. 

23.  Find  the  sura  of  19,  23a,  16a,  and  27a.       Su.n,  83». 

24.  Find  die  sum  of  At,  2\'^,  3»,  25a,  and  33a 

25.  Find  the  sum  of  12^,  18^,  33a,  87a,  and  93^ 

Sum,  245A 

26.  What  is  the  sum  of  »,  6a,  3|,  2a,  and  98  ? 

Ans.  111». 

27.  What  is  the  sum  of  1,  2a,  3a,  6a,  and  9a? 

79 


S134  ABSTRACT   NUMBERS. 


28.  AVhat  is  the  sum  of  4,^,  5-«,  17|,  and  18y^  ? 

An?.  46 1^. 

29.  What  is  the  sum  of  2i,  251,  125^,  and  325tV? 

Ans^  478/j-. 

30.  What  is  the  sum  of  If,  4|,  7«,  10-H-,  and  13-}-^? 


SUBTRACTION  OF  COMMON  FRACTIONS. 


Ex.  1.  From  -}  take  f. 

7_.3_4_i  §134.    M->DEL.— B  from   7  leaves  4. 

4  eighths  is  equal  to  -}.     The  reiuaiade? 


b  B  2 


IS^, 


Ex.  2.  From  i  take  i 


1 


§  135.  Model. — Keduce  the  fractions  to 
^  their  least  conimon   denominator.   (>?  loO). 

-^— |^=i     2  from  3  lenves  1,  that  is,  1  sixch.     The  re- 
mainder is  -}.. 


a' 


Ex.  3.  From  32?-  take  ISf. 

32|-         x_A  ^  1S().    Model.— Reduce   the 

•^Q?  y      g_^  fractions  to  their  least  common 

14i-         "B'~s  — 8  denomi»)ator.    (§130).     G  from 

7  leavt's  1,  set  down  |-;  8   from 
12  lenves  4  ;  2  from  3  leaves  1.     The  rcmaiuder  is  14}. 

ExiLANATiON. — Any  numher  of  fractional  units  may 
evidently  be  subtracted  from  a  larger  number  of  fractional 
units  of  the  same  denomination,  just  as  one  number  of  sim- 
ple units  is  subtracted  from  an  other.  If  the  given  frac- 
tions liave  different  denominators,  they  must  first  be  re- 
duced to  a  common  denominator  :  ^  — ^  =  neither  |  nor  ^ 
just  as  7  dollars  — 3  cents=neither  4  dollars  nor  4  cents. 
Ex.  4.  From  27  take  19}. 

27  §  137.  MoDKL. — 1  from  8  leaves  7,  set  down 

■L^i       -^r  10  from  17  leaves  7;   2  from  2  leaves  0. 
7l-      The  remainder  is  7|. 

80 


8" 


SUBTRACTION    OF   COMMON    FRACTIONS.  §138 

Ex.  5.  From  9^  take  6h 

^8         A  —i         §  138.  Moi^EL. — Reduce  the  fractions 

"2  to  their  least  common   denominator. — 

2-5.         8~-¥     (§  130).     4  from  9  leaves  5,  set  down  ^ 

7  from  9  leaves  2.     The  remainder  is  2| 

Explanation. — When  the  fraction  in  the  minuend  is 
less  than  that  in  the  subtrahend,  we  add  an  integral  unit  to 
the  minuend  fraction,  subtract  the  subtrahend  fraction 
from  this  sum,  and  then  add  1  to  the  unit^  of  the  subtra- 
hend before  subtracting  from  the  units  of  the  minuend. 

Rule. — Reduce  the  fractions  to  their  least  common  de- 
nominator  ;  sid>tract  the  numerator  of  the  stibtraheQid  from 
the  numerator  of  the  minuend;  and  under  the  remainder 
set  the  common  denominator. 

Jf  111  suhtracting  one  mixed  number  from  an  other,  the 
suhtrcdicnd  fraction  should  he  larger  than  the  one  in  the  min- 
uend, reduce  an  integral  unit  to  the  common  denomination 
of  the  fractions,  add  if  to  the  minuend  fraction,  subtract  the 
subtrahend  fraction  from  this  sum,  and  add  one  to  the  sub- 
trahend in  the  column  of  units, 

Ex.  6.  Subtract  f  from  y^ 

7.  Subtract  I  from  ^.  Rem.  -^, 

8.  Subtract  -{j^  from  -^^.  Rem.  i. 

9.  Subtract  f  from  -Li. 

10.  Subtract  -f-  from  |.  Rem.  -i-f . 

11.  From  \-l  take  «.  Rem.  ^. 

1 2.  From  -}/^  take  |. 

13.  From  5§-  take  f.  Rem.  5^- 

14.  From  7^  take  4^.  Rem.  S^^-. 

15.  From  8f-  take  7f , 

16.  Minuend=l7-iV;  Subtrahend=6i  Rem.  lOff. 

F  81 


§139  ABSTRACT    NUMBERS. 


17.  Minuend:=2003^;  Subtrahend=105^.  Rem.  94| 

18.  Minuend  =:42i;  Subtrabend  =  27yV 

19.  Minuends  721;  Subtrahend =24fV-  Rem.  47| 

20.  Minuend=l75;  Subtrahend  =:83|-.  Rem.  91-1; 

21.  Subtrahend =661-;  Minuend=106^ 

22.  Subtrahend  =  171;  Minuend  =  27i  Rem.  10/, 

23.  Subtrahend=lf ;  Minuend=4.f.  Rem.  3| 

24.  Subtrahends  7-«-;  Minuend=8y%. 

25.  Subtrahend  11-}^  ;  Minuend=20f5-.  Rem.  8f|f 

26.  What  is  the  difference  between  12|i  and  21ii? 

Ans.  8-}^ 

27.  What  is  the  difference  between  16-^-|^  and  10-JvV? 

28.  What  is  the  difference  between  100  and  33^? 

Ans.  66| 

29.  What  is  the  difference  betwcc;.  :  9}  and  20yV? 

Ans.  I-I- 

30.  What  is  the  difference  between  75  and  68^-1 


MULTIPLIOATION  OF  COMMON  FRA^CTIONS. 


Ex.  1.  Multiply  I  by  7. 

§  139.  Model. — 7  times  3  are  21 : 
Ax  7=  V=2|     21  eighths  is  equal  to  2|,     The  prod- 
uct is  2-^r. 

Explanation. — Comparing  §§  72  and  111,  we  see  that 
the  value  of  a  fraction  is  multiplied  by  a  whole  number  by 
multiplying  its  numerator  by  the  number. 

Ex.  2.  Multiply  |-  by  3. 

§  140.  Model. — 3  in  9,  3  times :  5 
Ax3=f=l|-    thirds  is  equal  to  If.     The  product  is 


1^ 


82 


MULTI1>LIGATI0N   OF   COMMON    FRACTIONS.  §143 

-  Explanation, — Comparing  §§  75  and  111^  we  see  that 
the  value  of  a  fraction  is  multiplied  by  a  whole  number  by 
dividing  its  denominator  by  the  number.  When  the  mul- 
tiplier is  a  measure  of  the  denominator,  this  method  if 
preferable  to  the  other. 

Ex.3.  Multiply  47|- by  9. 

^'^4:  §  141.  Model.— 9  times  3  are  27  :  #27  fourths 

^  is  equal  to  6-2-,  set  down  a  •  9  times  7  are  63,  and 

429f  G  are  69,  set  down  9  ;.  9  times  4  are   36,  and  6 

■  ^  are  42.     Tbe  product  is  429-2. 

Explanation. — As  in  whole  numbers,  we  begin  with  the 
lowest  denomination,  and  reduce  each  partial  product  to 
the  next  higher  denomination,  setting  down  the  remaining 
units  of  the  denomination  in  question,  and  reserving  the 
units  of  the  next  denomination  to  be  added  to  the  nest 
product. 

Ex.  4.  Multiply  ;]  by  f . 

3     7_2_i         §142.  Model.— 7  times   3   are   21:    S 
4.  X  s  — 3-z     times  4  are  32..     The  product  is  fi. 

Explanation. — To  multiply  by  |-  is  the  same  as  to  mul- 
tiply by  7  and  divide  the  product  by  8.  7  times  3  fourths 
=:21  fourths,  and  21  fourths-^8  =  21  thirty-seconds:  since 
a  fraction  (or  a  quotient)  is  divided  by  multiplying  the  de- 
nominator (or  the  divisor).     (§§  74,  111). 

Ex.  5.  Multiply  30,^  by  l. 

op.1     ^  §143.  Model.— Reduce  30i  to 

^^^" ^  *'  fourths.(§l26).   It  is  equal  to  121 

i|-i  X  i=  VV  =7-5%^     fourths.  Once  121  is  121 :  4  times 

4  are  16.     121  sixteenths  is  equal 
to  7^.     The  product  is  7^\. 

83 


U44  ABSTRACT    NUMBERS. 


Ex.  6.  Multiply  30^  by  5^ 

30i  X  6i-  ^  ^'^^'  ^^^^^^- — I'^educe  the 

*        ^  mixed  numbers  to  improper 

i|i  X  V  =  i-^/J  =  166f     fraction?.    (§  126).     11  times 

121  are  1831 :  twice  4  are  8, 
1331  eighths  is  equal  to  166^.     The  product  is  166|. 

Explanation. — It  is  often  easier  to  reduce  a  mixed 
iiumber  to'Hin  improper  fraction  before  multiplying,  if  the 
other  factor  is  not  a  whole  number. 

Rule. — To  multiply  a  simple  fraction  by  a  whole  number; 

Divide  the  denominator  of  the  fraction,  or  else  multiply 
lis  numerator^  by  the  whole  number. 

To  multiply  a  fraction  by  a  fraction. 
Multiply  each  term  of  the  one  fraction  by  the  correspond- 
ing term  of  the  other. 

A  mixed  number  may  be  reduced  to  an  improper  frac- 
tion, or  its  parts  may  be  multiplied  separately. 
Ex.  7.  Reduce  |  of  4  to  a  simple  fraction. 

„      .     \  §  145.  Model. — Twice  4  are  8  :  3  times 

'3      5"     i'5     5  Q^pQ  1^    fj^i^Q  given  fraction  is  equal  to -^^g. 

Explanation. — One  third  of  1  fifth  is  evidently  1  fif- 
teenth ;  1  third  of  4  fifths  is  4  times  1  fifteenth,  that  is,  4 
fifteenths  ;  and  2  thirds  of  4  fifths  is  twice  4  fifteenths,  that 
is,  8  fifteenths. 

Rule  for  reducing  a  compound  fraction  to  a  simple  one. 
Multiply  together  the  several  fractions  ivhich  compose  it. 
Ex.  8.  Multiply  a  by  4.  Prod.  1^. 

9.  Multiply  A  by  7. 

10.  Multiply  -l  by  8.  Prod.  7. 

11.  Multiply  I  by  12.  Prod.  10. 

12.  Multiply  yV  by  15. 


MULTIPLICATION    OF   COMMON    TRACTIONS.  §145 


13.  Multiply  -j^o  by  5.  Prod.  ^. 

14.  Multiply  2i  by  7.  Prou.  17l. 

15.  Multiply  8-1  by  8. 

16.  Multiply  16f  by  15.  Prod.  250. 

17.  Multiply  19^.  by  20.  Prod.  397^. 

18.  Multiply  207-;}  by  13. 

19.  What  is  the  product  of  315-^  and  19  ?        Prod.  5995|. 

20.  What  is  the  product  of  -'■-  and  -•}  1  'Prod.  f-. 

21.  What  is  the  product  of  |  and  a? 

22.  AYhat  is  the  product  of  J^  and  ^5- 1  Prod.  -}. 

23.  Reduce  ^  of  §-  to  a  simple  fraction.  Val.  -*. 

24.  Keduce  jj  of  ;^  to  a  simple  fraction. 

25.  Reduce  -^-  of -f  to  a  simple  fraction.  Val.  -f-l. 

26.  Reduce  -}  of  ^l  of  ^  to  a  simple  fraction.  Val.  ■^. 
'11 .  Reduce  -i-  of  -f  of  -^^  to  a  simple  fraction. 

28.  Reduce  -^  of  7-^-  to  a  simple  fraction.  Val.  2-}. 

29.  Reduce  \-  of  \  of  7^  to  a  simple  fraction.  Val.  l^-. 

30.  Reduce  -";  of  -^  of  8^  to  a  simple  fraction. 

ol.  Find  the  product  of  f  of  f  and  \:  of  12? .        Prod.  1-^^. 
32.  Find  the  product  of  f  of  f  and  83^-.  Prod.  2^\, 

83.  Find  the  product  of  -^-  of  66f  and  -I  of  100. 

34.  Find  the  product  of  f  of  250  and  -f  of  21.      Prod.  900. 

35.  Find  the  product  of  -\  of  ^  of  210  and  \  of  83|. 

Prod.  670^. 
SO.  What  is  the  product  of  16|  and  16^? 

37.  What  is  the  product  of  30^  and  60|-  ?  Ans.  183a^. 

38.  What  is  the  product  of  111-,^^  and  20-^  ?     Ans.  2277^^. 

39.  What  is  the  product  of  275  and  "^  of  a  of  36? 

40.  What  is  the  product  of  303  and  |-of  20  ?     Ans.  173^-. 

41.  What  is  the  product  of  3|-  and  4^4? 

42.  What  is  the  product  of  ^  of  f  and  f  of  3f  ? 


§146  ABSTRACT    NUMBERS. 


DIVISION  OF  COMMON  FRACfTIONS. 


Ex.1.  Divide  i-^.  by  3. 
jL^_i.3— _5_        §  ^"^^^  Model.— 3  in  15,  5  times.    The 


^'^  '         ^°     quotient  is 


1  6' 


Explanation. — Comparing  §§  73  and  111,  we  see  that 
the  value  of  a  fraction  is  divided  by  a  whole  numbet*  by 
dividing  its  numerator  by  the  number. 

Ex.  2.  Divide  f  by  5. 


K 3 

■^— To 


§  147.  Model.— 5  times  4  are  20.     The 
quotient  is  3  twentieths. 
Explanation. — Comparing  §§  74  and  111,  we  see  that 
the  value  of  a  fraction  is  divided  by  a  whole  number  by 
multiplying  its  denominator  by  the  number. 
Ex.  3.  Divide  f^  by  |. 

§148.  Model. — 3  in  15,  5  times: 
-ff-f-f  =:;f=li     4  in  16,  4  times.     5  fourths  is  equal 
to  1|:.     The  quotient  is  l-i. 
Explanation. — 15  sixteenths-^3=i5  sixteenths  (§146): 
but  the  divisor  3  fourths  is  only  one  fourth  of  3  ;  hence  the 
quotient  is  4  times  5  sixteenths,  that  is,  5  fourths.  (§  140). 
Again,  since  division  is  the  reverse  of  multiplication,  the 
process  for  division  should  be  the  reverse  of  that  for  multi- 
plication :  and  since  -|-x-f-=:-j^-g^,  it  is  evident  that  -^^-^2^=-^. 
Ex.  4.  Divide  |  by  |. 

§  149.  Model.— 8  times  3  are  24  :  7 
.l-hf ?=ff =-f    times  4  are  28.     24   twenty-eighths  ia 
equal  to  f.     The  quotient  is  ^. 
ExPIiANATiON. — 3  fourths-^7=:3  twenty-eighths  (§147): 
but  the  divisor  7  eighths  is  only  one  eighth  of  7  ;  hence,  by 
§75,  the  quotient  is  8  times  3  twenty-eighths,  that  is,  24 
twenty- eighths.  (§  139). 

86 


DIVISION   OF   COMMON   FRACTIONS.  §151 


Again,  Multiplying  both  terms  of  the  dividend  by   56, 
we  have  a||-^|^= I4=t-     ^^>  Multiplying  both  terms  by 
14,  we  have  -;|^f -^|-=-f,  the  same  result  as  before. 
Ex.  5.  Divide  2731-  by  5. 

§  150.  Model. — 5  in  27,  5  times  with  2 
l?_i     over,  set  down  5 ;    5  in  23,  4  times  with   3 
54f     over,  set  down  4;  5  in  10,  twice,  set  down  f. 
The  quotient  is  54f. 
Explanation. — We   divide  the  integer  as  usual,  and 
reduce  the  3  remaining  units  to  thirds,   making  9  thirds, 
which  added  to  the  given  1  third  makes  10  thirds,  and  this 
divided  by  5  gives  2  thirds.     If  the   numerator  of  \£  had 
not  been  divisible  by  5,  we  would  have  multiplied  its   de- 
nominator by  the  divisor,  as  in  §  147. 
Ex.  G.  Divide  3^-  by  12f . 

^^  .193  §151-    Model. — Reduce   the 

3""   *-'3  given  mixed  numbers  to  improp- 

i_o_^.3.s__3^__5_        QY  fractions.    (§126).     3   times 

10  are  30  :  38  times  3  are  114, 
30  one-hundred-and-fourteenths  is  equal  to  5  nineteenths. 
The  quotient  is  5  nineteenths, 

KuLE.— -To  divide  a  simple  fraction  by  a  whole  number. 
Divide  the  numerator  of  the  fraction^  or  else  multiply  its 
denominator  J  by  the  tchole  number. 
To  divide  a  fraction  by  a  fraction. 

Divide  each  term  of  the  dividend  by  the  correspo/iding 
term  of  the  divisor.  Or,  Multiply  each  term  of  the  divi- 
dend by  the  other  term  of  the  divisor. 

To  divide  a  whole  number  by  a  fraction. 
Divide  the  dividend  by  the  denominator  of  the  divisor, 
(t.nd  multiply  the  quotient  by  the  numerator . 

A  mixed  number  will  mostly  better  be  reduced  to  an  im- 
proper fraction. 

87 


§162 


ABSTRACT    NUMBERS. 


Ex.  7.  Reduce 


2i 


of 


to  a  simple  fraction. 


2i-^iof  A 


40. 

6    ' 


§  152.  Model. — Reduce  the  terms 
to  simple  fractions.  Divide  |-  hy  ^. 
8  times  5  are  40  :  3  times  2  are  6. — 
40  sixths  is  equal  to  6f .  The 
fraction  is  equal  to  6|. 

Rule  for  reducing  a  complex  fraction  to  a  simple  one. 
Divide  its  numerator  by  its  denominator. 
Ex.  8.  Divide  f  by  5. 
9.  Divide  -^  by  8. 

10.  Divide  if  by  3 

11.  Divide  if  by  6 

12.  Divide  40  by  f 

13.  Divide  200  by 

14.  Divide  175  by 

15.  Divide  \%  by  | 
10.  Divide  |-^-  by  f. 

17.  Divide  i|  by  f . 

18.  Dividend=-^' 

19.  Dividend^! 

20.  Dividend=f 

21.  Dividend=:y^ 

22.  Divisor =f 

23.  Divisor^l 

24.  Divisor={r 

25.  Divisors  ^ 

26.  Divisorz^i  off: 

27.  Divisors  I  of  f  : 

28.  Divisor=:f  of  f  : 

29.  Divisor=f  of  l  : 

30.  Divisor=i  of  12^ 


divisor =-|. 
4 :    divisor=|-. 
divisor=:f. 
divisor=i. 


dividend =-^. 
dividend=^|. 
dividend ={'. 
dividend  =.^. 

dividend  =  |. 

dividend=-f-. 

dividend  =  *-  of  ^. 

dividend=f  of  ^0-.. 

:    dividend  =  1  of  y\ 

88 


given 


Quot.  ^V 

Quot.  -V 
Quot.  -^-^. 

Quot.  466|-. 
Quot.  49. 

Quot.  ia. 
Quot.  |. 

Quot.  2^Q, 
Quot.  i'A. 

Quot.  li 
Quot.  ^^, 

Quot.  ,«. 
Quot.  Iff. 

Quot.  ^. 
Quot;  1\%. 


DIVIvSION   or    COMMON   FRACTIONS.  §152 


:;l.  Dividend=:12A-:    clivisor=4.  Quot.  3^. 

:i2.  Dividend:rz207i  :    divisor^G.  Quot.  S^-^V 

:;3.  DivideDd=i45f :    divisor^lS^. 

34.  Dividends 70^:    divisor=68^.  Quot.  l-^-f^. 

35.  Dividends 27-^:    divisor=:55i            '  Quot.  |-f|. 

36.  Dividend^:^  of  28^  :    divisori=:|  of  43^}. 

37.  Dividend=:|-  of  a  ;    divisor=:i  of  275.  Quot.  ■^^^. 
:]8.  Dividends 2  of  i  of  I- :    divisor=|  of  17-^-.  Quot.  ■^. 

39.  Dividend=|  of  27.V:    divisor=^  of  38^. 

40.  Pteduce  — "-  to  a,  simple  fraction.  Val.  3-|. 

4." 

41.  Reduce  -^  to  a  simple  fraction.  Val.  --^y\. 

4r2.  Reduce  — ^  to  a  simple  fraction. 
41 

^r  of  ? 

43.  Reduce  "       "  to  a  simple  fraction.  Yal.  ■^\. 

2,  of  4^ 

44.  Reduce  ~ — — -^  to  a  simple  fraction.  Yal.  i|-. 

-}  ot  19 

'}.  of  ?- 

45.  Reduce  --—7-"-  to  a  simple  fraction.. 

I  oi  7i  ^ 

27 

46.  Reduce  — -.— — -  to  a  simple  fraction,  Val.  2tV- 

■I  of  30  ^  ' "" 

1-  of  20 

47.  Reduce  — — — -  -  to  a  simple  fraction.  Val.  -;;%• 

-|ofl7-i-  ^ 

?  of  2^- 

48.  Reduce to  a  :  iinpio  fraction, 

3461 

2i- 
40.  Reduce  ~  to  a  simple  fraction.  Val.  1- 

1  of  27j 
50.  Reduce  - — rr~  to  a  simple  fraction.  Val.  WA\« 

f-of72f  "  '''^^' 

89 


^153  ABSTRACT   NUMBEHS. 


CANCELLATION. 


§  153.  In  multiplication  of  fractions,  and  in  some  other 
similar  operations,  the  labor  may  be  often  diminished  by 
canceling  all  the  factors  common  to  the  numerators  and  the 
denominators,  and  afterwards  multiplying  together  the  re- 
maining factors  of  each.  This  is  simply  reducing  the  re- 
sult to  its  lowest  terms  in  advance. 

It  is  customary  to  draw  a  line  through  a  number  that  has 
"been  canceled. 

Ex.  1.  Multiply  U  by  Ti^. 

^     7__^         Model. —  45  in  45,  once;   45  in  90, 
f^'  pfp"~14     twice:    7  in  7,  once;    7  in  49,  7  times  :- 
7       2  the  numerator  is  1 ;  the  denominator  is 

7x2  =  14.     The  product  is  ^L. 

Ex.  2.  Divide  I-  of  j-  of  f  by  f  of  if  of  ^V- 

i  of  f  of  i---f  of  i|  of  ^\  Model.— 3 

^  in  3,  once  ;    3 

-  of  2  of  '?  X  ^  of  ?^  of  —  =..  «J-  =  10-^-     ^^  ^'  ^^^^®  •  ^ 
27^^^/^         1         «         ^      in7,  once;7 

2         2  in  21, 3  times: 

9  in  9,  once  ; 
9  in  9,  once  :  5  in  5,  once ;  5  in  10,  twice  :  the  numerator 
is  3  X  27=81  ;  the  denominator  is  2.2.2=8.  The  quotient 
isV  =  10i. 

Ex.  3.  Divide  the  product  of  77  and  96  by  the  product 
of  22  and  24. 

2  Model.—  11  in  77,  7  times  ;  J 1  in  22^ 

^     /¥  twice  :   2  in  2,  once  ;    2  in  96,  48  times  • 

Ilj^.^14,  24  in  24,  once;    24  in  48,  twice.     The 

^^  X  ^4'       ^  quotient  is  7  x  2=  14. 


? 


90 


PROMISCUOUS   PP.OBLEMS.  SI 53 


Ex.  4.  Divide  11  x  21  x  26  by  3  x  13  x  14. 

^^^Jy^^li        Model.—  3  in  3,  once;   3  in  21,  7 
11  X  //^xjg     times  :  7  in  7,  once  ;  7  in  14,  twice  :  13 
^x/^x/^     in  26,  twice;   13  in  13,  once:    2  in  2, 
^     once ;  2  in  2,  once.    The  quotient  is  11 
Es.  5.  Multiply  A  of  i-f-  by  -J^  of  f  ^^  of  if.         Prod.  ^ 

6.  Multiply  -V  of  I-  of  if  by  I  of  -y-. 

7.  Multiply  f  of  V  by  a  of  if.    "  Prod.  ^'^ 

8.  Divide  ^  of  AA  by  V  of  V  of  -^V-  Quot.  i 

9.  Divide  f-t  of  if  by  -Jl.  of  ^. 

10.  Divide  A  of  A  of  I  by"A  of  aa.  Quot.  2i 

11.  Divide  the  product  of  22  and  56  by  the  product  of  44, 

28,  and  16.  Quot. -,„ 

12.  Divide  the  product  of  72  and  96  by  the  product  of  60 

and  64. 

13.  Divide  the  product  of  27,  2S,  and  29  by  the  product  of 

35,  86,  and  37.  Quot.  -/a.. 

14.  Divide  10  X  11  x  12  by  22  x  24  x  30.  Quot.' J-^. 

15.  Divide  25  x  27  x  32  x  36  by  15  x  18  x  24  x  28. 


PROMISCUOUS  PROBLEMS. 


1.  What  is  the  sum  of  275,  386,  497,  and  608  ? 

2.  What  is  the  difference  between  275386  and  497608  ? 

Ans.  222222. 

3.  What  is  the  product  of  275386497  and  608  ? 

4.  What  is  the  quotient  of  275386497  by  608  ? 

Ans.  452938J-AA. 
-J.  Add  the  difference  between  395  and  4:22  to  the  sum  of 
39,  54,  and  202.  .  Sum,  3 

91 


§163  ABSTRACT   NUMBERS. 


(>.  Subtract  the  sum  of  25  and  19  from  their  product. 

7.  Multiply  ibe  difference  of  25  and  19  by  tlieir  sum. 

Prod.  264.. 

8.  Divide  the  product  of  36  and  45  by  their  difference. 

Quot.  180. 

9.  Resolve  7050  into  its  prime  factors. 

10.  What  is  the  greatest  common  measure  of  25,  250,  and 

375  ?  "  Ans.  25. 

il.  What  is  the  least  common  multiple  of  5,  G,  10,  and  12  ? 

Ans.  60. 
1-.'.  Reduce  -rV-^V  to  its  lowest  terms. 

i  z  o  u 

13.  In  "^"^  how  many  units  ?  Ans.  10-^^. 

14.  In  19  units  how  many  nineteenths  ?  Ans.  ^^. 

15.  In  15|-  how  many  fifths  ? 

16.  In  f  how  many  forty-fifths  ?  Ans.  -|-|.. 

17.  In  -^q\  how  many  twenty-fifths  ?  Ans.  -^V- 

18.  Reduce  -^,  -^j  and  {-  to  a  common  denominator. 

19.  Reduce  ^,  |,  and  {^  to  their  least  common  denominator. 

Vol       2  7       .TO       2_5. 

20.  Add  -5^,  Jj,-,  and  i}.  Sum,  1. 

21.  What  is  the  sum  of  }  of  *-  and  ^  of  i  ? 

2^ 

22.  What  is  the  sum  of  f  of  10-^  and  — -^— ,  ?    Ans.  7/,%. 

•"  "*  i  of  17 

23.  What  is  the  difference  between  19^-  and  26^  ? 

Ans.  6-1-^. 

24.  What  is  the  difference  between  a  of  27  and  -^-  of  24? 

25.  What  is  the  product  of  27i.and  -v  of  77  ?      Ans.  706|-. 

26.  What  is  the  product  of  '^-—^rr- — -  and  — ---   '/     Ans.  1. 

^  22^  .}  or  11 

27.  What  is  the  quotient  of  I  of  47  by  25^  ? 

28.  What  is  tlie  product  of  5-  of  47  and  ^  of  25  ? 

Ans.  43-1-*-. 
93 


I'LlOiVllSCUO'tJS   PROBLEMS.  §l5o 

-I-  of  27.^        ^  of  19  , 
29.  Wlhit  h  the  quotient  of  ^—xz — -  by  -—     ^    ■  ? 

Aids.  3,W.V 
SO.  Add  the  product  of  -^  of  27  and  ~l  of  ^  to  their  difference. 

31.  Subtract  the  quotient  of  -f-  of  45  by  -}  of  24  from  their 
sum.  Eem.  lO^'j. 

A  of  5- 

32.  Multiply  the  sum  of  ^"        "    and  ?  of  7'Sl  by  their  dit- 

ference.  Prod.  262-jyVc- 

33.  Divide  the  product  of  25^-  and  17^,-  by  their  sum. 

34.  What  number  is  that  to  which  if  3-§- ,  5^^-,  6f,  and  10-/^, 
be  added,  the  sum  will  be  30-]j-  ?  Ans.  3-^-. 

35.  What  number  is  that  tVoni  which  if  of,  of,  6^^,  and  10^%, 
be  subtracted,  the  remainder  will  be  30-^  ?     Ans.  57-|-. 

36.  What  number  is  that  by  which  if  the   sum   of  3f,  5|^ 
6{?,  and  10y%,  be  multiplied,  the  product  will  be  30-|-? 

37.  What  number  is  that  by  which  if  the   sum  of  3f,  5f, 

6|,  and  lOy'^;-,  be  divided,  the  quotient  will  be  30-1-  ? 

Ans  -^-i- 
88.  What  is  the  sum  of  -},  ,|,  13,  and  IS^V  ? 

39.  What  is  the  difference  between  -}  and  -^\1         Ans.  y%. 

40.  What  is  the  product  of  5^^  and  ^  ? 

41.  What  is  the  quotient  of  -ff  ff  by  19  ? 

42.  3^-5-7-^5  +  16-^7-15'^6=what?  Ans.  6|-^. 

43.  (3  +  5-7)~5  +  16-^7-15-=-6=what? 

44.  3  +  (5  +  7)~5-fl6-^7-15-J-6=what?  Ans.  5-^-§.. 

45.  3  +  5-7-^6-fl6  +  7-15-^6=what?  Ans.  27yV- 

46.  (3  +  5-7)-^-5  +  (16  +  15-7)-f-6=what? 

47.  3  +  (5  +  7)-^5  +  16+(15-7)-^6=:what?      Ans.  22H-. 

48.  3  +  (7-5)-^(5  +  16)  +  (15-7)-^6=what?        Ans.  4f 

49.  34.7_5^(5_l_16  +  15)-7-f-6=what? 

93 


§154  ABSTRACT    NUMBERS. 


DECIMAL  FRACTIONS. 


§  154.  A  decimal  fraction  is  one  whose  denominator  is 
some  poivcr  of  ten  and  is  not  expressed  in  writing. 

§  155.  In  the  Arabic  or  decimal  system  of  notation  (§10), 
we  observed  that,  in  passing  from  the  units'  place  to  the 
left,  a  unit  of  any  order  is  ten  times  a  unit  of  the  preceding 
order ;  or  that,  in  passing  from  left  to  right,  a  unit  of  any 
order  is  07ie  tenth  of  a  unit  of  the  preceding  order.  If  this 
law  be  extended  to  the  right  of  units,  the  next  order  will 
be  tenths,  the  next  hundredths,  the  next  thousandths,  &c., 
as  in  the  following 


■  .  ^  ti  -a 

^  »        I     t2 1  S  r  2  S  S  ^  i  ^  ^^ 

23  45.234576595875 

As  100  is  3-V  of  1000, 10  is  3-V  of  100,  and  1  is  J^  of  10, 
so  one  tenth  is  -^  of  1,  one  hundredth  is  -=^^  of  1  tenth,  one 
thousandth  is  J^  of  one  hundredth,  &c. 

§  156.  To  write  any  number  of  tenths,  then,  we  simply 
put  the  proper  figure  one  place  to  the  right  of  units ;  for 
hundredths,  we  put  the  figure  two  places  to  the  right,  &c. 
To  determine  the  position  of  units  and  the  relative  posi- 
tions of  the  fractional  orders,  we  place  a  period,  called  the 

94 


NOTATION    OF   DECIMAL   FRACTIONS.  §158 


units^  point,  between  units  and  tenths ;  or  to  the  left  of 
tenths,  if  the  expression  is  entirely  fractional.  Thus,  2.3, 
two  and  three  tenths  ;  3.02,  three  and  two  hundredths ; 
5.32,  five  and  three  tenths  and  two  hundredths ;  .005,  five 
thousandths ;  .0006,  six  ten-thousandths ;  .00004,  four 
hundred-thousandths;  .000008,  eight  milliontht-. 

In  integral  numbers,  this  point,  being  unnecessary,  is 
never  written  :  but  in  fractional  or  mixed  expressions,  it 
must  never  be  omitted. 

§  157.  It  will  be  observed  that  the  number  of  places  oc- 
cupied by  the  numerator  of  a  decimal  fraction  is  equal  to 
the  number  of  naughts  in  its  denominator.  If  the  ordinary 
exprei?sion  cf  the  numerator  does  not  require  so  many  places, 
each  place  intervening  between  the  units'  point  and  the  left 
hand  figure  of  the  numerator  must  be  filled  with  a  naught. 
Thus,  .002,  2  thousandths;  .023,  23  thousandths ;  ,0203, 
203  ten-thousandths  ;  .0023,  23  ten-thousandths ;  .0004,  4 
ten-thousandths  ;  .002034,  2034  millionths. 

§  158.  A  decimal  fraction  is  read,  like  a  common  frac- 
tion, by  pronouncing  after  the  numerator  the  ordinal  of  the 
denominator.  Sometimes,  in  reading  a  mixed  number,  to 
prevent  ambiguity,  it  is  necessary  to  pronounce  the  word 
"units"  after  the  integer.  Thus,  three  hundred  and  fif- 
teen thousandths  is  written  .315  ;  but  300.015  is  three  hun- 
dred units  and  fifteen  thousandths :  so,  7000.0275  is  read 
seven  thousand  units  and  two  hundred  and  seventy-five  ten- 
thousandths. 

Read  the  following  decimal  fractions: —  .1,  .3,  .5,  .7,  .8  • 
.01,  .05,  .09,  .11,  .25,  .34,  .47,  .51,  .63,  .75,  .87,  .99;  .001^ 
.005,  .015,  .025,  .075,  .125,  .219,  .375,  .487,  .567,  .605,  .777, 
.808,  .999;   .0001,  .0012,  .0125,  .1275,  .3525,  .6225,  .7203, 

95 


§158  ABSTRACT   mTMBERS. 


.8007,  .9883,  .9999 ;  .00001,  .00014,  .00225,  .03275,  .33125, 
.42075,  .53003,  .70007,  .87078,  .99999 ;  .000001,  .000017, 
.000175,  .003175,  .063175,  .475327,  .796305,  .634008, 
.320075,  .200017,  .200325;  .0000001,  .0000025,  .0000275, 
.0020705,  .0357675,  .7500786;  .00027625,  .02700625, 
.23450275,  .00073513,  .23570025,  .125346798,  .000000125, 
.0007600025,  .27340709025,  .70030005345,  .000257025702. 


Read  the  following  mixed  numbers : —  3.3,  70.5,  35.7, 
2.02,  3.25,  75.75,  24.05,  7.07,  30.003,  400.025,  25.125, 
375.375,  1.001,  2.325,  2.0275,  300.0025,  17.0017,  1.0005, 
2000.0002,  21.2125,  325.03725,  9180.20025,  1000.02207, 
7025.00025,  6278.374375,  2000.0002325,  3375.00000765, 
27.0000027,  3200.000000075,  2500,0000036975. 

Write  tlie  following  in  figures  : 

110.  Seventy-four  hundredths. 

111.  Four  hundred  and  forty-eight  thousandths. 

112.  Five  hundred  units  and  three  hundredths^. 

113.  Seventy-five  thousandths. 

114.  Five  hundred  and  three  thousandths. 

115.  Five  hundred  units  and  three  thousandths. 

116.  Three  hundred  and  twenty-seven  ten-thousandths. 

117.  Three  hundred  units  and  twenty-seven  ten-thousandths. 

118.  Seventeen  and  seventeen  hundred-thousandths. 

119.  One  thousand  units  and  two  thousand  two   hundred 
and  seven  hundred-thousandths. 

120.  Three  thousand  two   hundred  units  and  seventy-five 
millionths. 

121.  Six  hundred  and  three  ten-thousandths. 

122.  Two  thousand  four  hundred  and  sixty-one  and  three 
hundred  and  nineteen  millionths. 

96 


ADDITION    OF   DECIMAL   FRACTIONS.  §160 


ADDITION  OF  DECIMAL  FRACTIONS. 


Ex.  1.  Add  .3,  .23,  .175,  and  .025. 

3 

*23  §  159.  Model. — 5  and  5  are  ^0  ;    1  and  2 

175         ^'^^  ^'  ^^^  ^  ^^^  ^^'  ^^^  ^  ^^^  ^^'  ^^^  down  3  ; 

025         ^  ^"^  ^  ^^®  "'  ^^'^  ^  ^^'^  ^>  ^^^  '^   ^^^  ^' — 
'-—         Point  before  7.     The  sum  is  .73. 

Explanation. — Beginning  at  the  right,  we  find  the  sum 
of  the  first  column  to  be  10  thousandths,  equal  to  1  hun- 
dredth exactly.  We  do  not  set  down  the  naught  here,  be- 
cause a  naught  at  the  right  of  a  decimal  fraction  does  not 
issist  in  determining  the  orders  of  the  other  figures.  The 
1  hundredth  is  added  in  with  the  column  of  hundredths, 
which  amounts  to  13  hundredths,  equal  to  1  tenth  and  3 
hundredths.  Setting  o  under  the  column  of  hundredths, 
we  add  the  1  tenth  in  with  the  column  of  tenths.  We  then 
place  the  units'  point  at  the  left  of  the  tenths.    (§  156). 

Ex.  2.  Add  .3,  3.5,  3.15,  35.25,  and  171.275. 

§  160.  Model.— 5  ;  7  and  5  are  12,  and 

•'^  5  are  17,  set  down  7 ;  1  and  2  are  3,  and  2 

'z'^  are  5,  and  1  are  6,  and  5  are  11,  and  3  are 

oi'ot  -^-^J  ^^^  down  4  ;  1  and  1  are  2,  and  5  are  7, 

171  07f^         and  3  ar6  10,  and  3  are  13,  set  down  3  ;    1 

^'^'^'^         and  7  are  8,  and  3  are  11,  set  down  1 ;    1 

213.475         and  1  are  2.     Point  before  4.     The  sum  is 

213.475. 

Explanation. — The  sum  of  the  column  of  tenths  being 
14,  that  is,  1  unit  and  4  tenths,  we  set  4  under  the  column 
of  tenths,  and  add  1  to  the  column  of  units.  We  place  the 
units'  point  between  the  units  and  the  tenths.    (§  156). 

a  97 


§160  ABSTRACT   NUMBERS. 


Rule. — Arrange  the  numbers  with  units  of  the  same  order 
in  the  same  column  ;  and  add  as  in  luholc  numbers.  (§  22). 

Place  the  units^  point  on  the  left  of  the  tenths  figure  in 
the  sum. 

Proof. — The  same  as  in  whole  numbers,  (§  22). 

Ex.  3.  Add  1.2,  3.56,  45.67,  and  56.789. 

4.  Add  1.3,  5.79,  24.68,  and  90.275.  Sum,  122.045. 

5.  Add  27.72,  365.9,  125.008,  and  236.115. 

Sum,  754.743. 

6.  Add  135.709,  246.008,  145.008,  and  236.709. 

7.  Add  1.35795,  135.795,  and  13579.5. 

8.  Find  the  sum  of  2.465,  25.G09,  100.206,  and  146.27. 

Sum,  333.950. 

9.  Find  the  sum  of  100.0001,  L- 1.4012,  412.5124,  and 

421.5214. 

10.  Find  the  sum  of  1234.58,78.9012,3456.789, 10.234567, 

and  890.13575.  Sum,  5670.020517. 

11.  Find  the  sum  of  907.0503,  890.7054,  785.4321,  and 

25.457.  Sum,  2608.6448. 

12.  Find  the  sum  of  12.012575,  120.125725,  1201.257725, 

and  .270825. 

13.  Find   the   sum   of   .760027,    .000176,    .012012,    and 

.027945.  Sum,  .800160. 

14.  What  is  the  sum  of  .230495,  .341507,  .452618,  and 

.563729  ?  Ans.  1.588349. 

15.  What  is  the  sum  of  2.30495,  34.1507,  452.618,  and 
5637.029  ? 

16.  What  is  the  sum  of  12.000012,  250.0025,  75.075,  and 

175.0175  1  Ans.  512.095012. 

17.  175  +  6.115  +  123.1341  +  172.21275  +  5637.175:^  what? 

Ans.  6113.63685. 
98 


SUBTRACTION    OF    DECIMAL   FRACTIONS.  §161 


18.  52.8G72  +  549.72-f927.365  +  57.10715  +  13.575=wliat? 

19.  79.105  +  131.187  +  19.4201 +2643.13  +  34.8H6-t  =  whai? 

Ans.  2907.6785. 

50.  3844.04  +  .444584  +  6.14644  +  6847.34  +  77.9899= 

what  ?  Ans.  10775.960924. 


SUBTRACTION  OP  DECIMAL  FRACTIONS. 


Ex.  1.  From  275.075  take  87.1275. 

^P'^I--         §  1^1-  Model.— 5  from  10  leaves  5  ;   8 

"'•  ^-'^^     from  15  leaves  7  ;    3  from   7   leaves  4  ;    1 

187.9175     from  10  leaves  9 ;  8  from  15   leaves  7;    9t 

from  17  leaves  8  ;  1  from  2  leaves  1.  Poiut 

before  9.     The  remainder  is  187.9475. 

Explanation. — After  placing  the  subtrahend  under  the 
minuend  with  units  of  the  same  order  in  the  same  column, 
we  find  5  ten-thousandths  in  the  subtrahend  and  no  ten- 
thousandths  in  the  minuend.  Adding  1  thousandth,  that 
is,  10  ten-thousandths,  to  the  minuend,  we  subtract  from 
this  the  5  ten-thousandths  of  the  subtrahend.  Then,  be- 
cause the  minuen'd  is  increased  10  ten-thousandths  or  1 
thousandth,  the  subtrahend  must  be  increased  the  same 
amount.  (§28).  The  same  kind  of  reasoning  will  explain 
the  rest  of  the  operation.  We  place  the  units'  point  be- 
tween the  units  and  the  tenths.    (§  156). 

Rule. — I^lace  ike  subtrahend  under  (he  minuend,  with 
*units  of  the  same  order  in  the  same  column,  and  subtract  as 
In  whole  numbers.    (§30). 

Place  the  unita^  point  on  the  left  of  the  tenths  figure  in  the 
remainder.  (|^  156). 

Proof. — The  same  as  in  whole  numbers.  (§  30). 

99 


S161  ABSTRACT   JSUMBER.^. 


Ex.  2.  From  8.96  take  8.07.  Rem.  5.89. 

3.  From  2.719  4:ake  1.827. 

4.  From  97.8637  take  9.7863.  Rem.  88.0774. 

5.  Take  67.8902  from  896.454.  Hem.  828.5638. 

6.  Take  17.24937  from  1963.869. 

7.  Take  234.68579  from  6005.004.  Rem.  5770.31821, 

8.  Take  98.79789  from  99.000099.  Rem.  .202209. 

9.  Minuend  is  1284.567  ;    Subtrahend  is  .76542„ 

10.  Minuend  is  29017.05  ;   Subtrahend  is  10.8405. 

Rem.  29006.2095. 

11.  Minuend  is  2098.76  ;  Subtrahend  is  454.698. 

Rem.  1644.062. 

12.  Minuend  is  1201.257725  ;    Subtrahend  is  120.125575. 

13.  Subtrahend  is  .012095;    Minuend  is  .027945. 

Rem.  .01585. 

14.  Subtrahend  is  2.30495  ;    Minuend  is  34.1507. 

Rem.  31.84575. 

15.  Subtrahend  is  12.000012  ;   Minuend  is  250.0025. 

16.  Subtrahend  is  75.075  ;   Minuend  is  175.0175. 

Rem.  99.9425. 

17.  5637.175-I72.2l275=:what?  Ans.  5464.96225, 

18.  927.305-57.190715=what?  ' 

19.  What  is  the   difference  between  one    millionth,   and 
ninety-nine  thousandths  ?  Ans.  .098999. 

20.  What  is  the  difference  between  thirty-seven  billionths, 

and  one  hundred  and  eleven  thousandths  ? 

Ans.  .110999963. 

21.  What  is  the  difference  between  six  billionths,  and  nine 
hundred  and  ninety-nine  thousandths  ? 

22.  What  is  the  difference  between  three  millionths,   and 
three  hundred  and  six  thousandths  ?        Ans.  .305997. 

100 


MULTIPLICATION   OF    DECIMAL   FRACTIONS.  §16S 


MULTIPLICATION  OF  DECIMAL  FRACTIONS. 


Ex.  1.  Multiply  5.3  by  6.25. 

6  25 
CO         §  162.  Model. — 3  times  5  are  15,  set  down 

— -—     5  ;    3  times  2  are  6,  and  1  are  7  ;    3  times  6 
^^1^     are  18: — 5  times  5  are  25,  set  down  5  under 
^^^^       7  ;  5  times  2  are  10,  and  2  are  12,  set  down  2 ; 
o3.125     5  times  6  are  30,  and  1  are  31: — add  the  par- 
tial products  :  5  ;  5  and  7  are  12,  set  down  2; 
1  and  2  are  3,  and  8  are  11,  set  down  1 ;    1   and   1  are   2, 
and  1  are  8 ;  3.     Point  before  1.     The  product  is  33.125. 
ExrLANATiON. — Reducing  both  factors  to  improper  frac- 
tions, and  multiplying  as  in  §  142,  we  have  -^-f^  x  {-^=  'VoW? 
and  this  product  reduced   to    a    mixed    number   becomes 
S3. 125.  as  in  the  model.     If  any  decimal  mixed  number  be 
reduced  to  an  improper  fraction,  the  numerator  will  con- 
sist of  the  same  figures  as  the  given  mixed  number.     Hence 
we  multiply  as   in   whole  numbers.     The   location   of  the 
units'  point  in  the  product  is  found  by  observing  that  the 
number  of  naughts  in  the  denominator  of  either  factor  is 
the  same  as  the  number  of  figures  in   the   numerator,  aijd 
that  the  product  of  any  two  powers  of  ten   is  obtained   by 
annexing  to  1  as  many  naughts  as  there  are  in  both  factors 
togetiier.     There  are,  therefore,  as  many  fractional  figures 
in  the  product  as  in  both  factors  together. 

Ex.  2.  Multiply  ,15  by  ,3, 

•^^         §  163.  Model. — 3  times  5  are  15,  set  down  5  ; 

3  times  1  are  3,  and  1  are  4.     Prefix  one  naught. 

.015     Point  before  0,     The  product  is  .045. 
Explanation. — When   the   product   does   not   contain 
onough  figures  to  express  its  proper  denomination,  we  pre- 
fix one  or  more  naughts  to  supply  this  deficiency. 

101 


>164  ABSTRACT    NUMBERS. 


Rule. — Midt^i-ily  as  in  loJioIe  numhers,  and  ^Doint  off  ai> 
many  fractional  Jigiires  in  the  product  an  there  arc  in  hotU 
the  facfon^y  prc/?x?*wr7  navghts  lohen  necessary  to  mahe  up 
the  number. 

Proof. — The  same  as  in  whole  numbers,  (§  40). 

Ex.  3.  Multiply  12.42  by  3.2. 

4.  Multiply  25.25  by  2.5.  ProJ.  63.125. 

5.  Multiply  .25  by  .25.  Prod.  .0625. 

6.  Multiply  5.5  by  5.5. 

7.  Multiply  211.79  by  2.7.  '    Prod.  571.833. 

8.  Multiply  97.825  by  .34.  Prod.  33.2605. 

9.  Multiply  275.005  by  5.005, 

10.  Multiply  869.06  by  .045.  Prod.  39.1077. 

11.  Multiply  27.9362  by  .0052.  Prod.  .14526824. 

12.  192.837x6.7==  what? 

13.  293.705  X  .075=what  ?  Ans.  22.027875. 

14.  3.047  x2.87=what?  Ans.  8.74489. 

15.  2.975  x.375=what? 

16.  4.027  X  402.7==  what  ?  Ans.  1621.6729- 

17.  What  is  the  product  of  247.742  and  10.035  ? 

18.  What  is  the  product  of  307.0005  and  .000375? 

19.  What  is  the  product  of  175.025  and  25.0175  ? 

Ans.  4378.6879375.* 
.20,  What  is  the  product  of  1200.375  and  162.625  ? 


DIVISION  OF  DECIMAL  FRACTIONS. 


Ex,  1.  Divide  2.25  by  .3. 

.3)2^         §  164.  Model.— 3  in  22,  7  'times,  with  1 
7.5     over,  set  down  7  ;   3  in  15,  5  times.     Point 
before  5.     The  quotient  is  7.5. 
102 


DIVISION  OF   DECIMAL   FRACTIONS. 


§166 


Explanation. — Since  the  divisor  and  the  quotient  arc 
factors  of  the  dividend,  there  must  be  as  many  fractional 
figures  in  the  dividend  as  there  are  in  both  the  factors. 
(§  162).  Hence,  to  find  the  number  of  fractional  figures  in 
the  quotient,  we  subtract  the  number  in  the  divisor  from 
the  number  in  the  dividend. 


26.40 
2475 


3.2 


1650 
1650 

0 


Ex.  2.  Divide  26.4  by  8.25. 

on-'  §165.  Model. — Annex  one  naught  to 
the  dividend  :  8  in  26,  3  times ;  multiply 
the  divisor  by  3  ;  15,  7,  24  ;  subtract  the 
product  from  the  dividend  ;  5, 6, 1 ;  annex 
0  :  8  in  16,  twice  ;  multiply  the  divisor  by 
2  ;  10,  5,  16 ;  subtract  the  product  from 
the  previous  remainder ;  0.  Point  before 
2.     The  quotient  is  3.2. 

Explanation. — As  the  number  of  fractional  figures  in 
the  divisor  exceeds  the  number  in  the  dividend,  we  annex 
a  naught  to  the  dividend  to  make  them  equal.  We  after- 
wards find  it  necessary  to  annex  an  other  naught  to  com- 
plete the  division.  This  makes  3  fractional  figures  in  the 
dividend  ;  and,  as  there  are  2  in  the  divisor,  there  must  be 
one  in  the  quotient. 

Ex.  3.  Divide  4  by  15. 


4.00 
30 


15 


.266  + 


100 
90 


100 
_90 

10 


§  166.  Model. — Annex  2  naughts  to 
the  dividend  ;  15  in  40,  twice  ;  multiply 
the  divisor  by  2  ;  10,  3  ;  subtract  the 
product  from  the  dividend  ;  0,  0,  1 ;  15 
in  100,  6  times;  multiply  the  divisor  by 
6  ;  30,  9  ;  subtract  the  product  from  the 
previous  remainder;  0,  1;  annex  1:  15 
in  100,  6  times  ;  &c.  Point  before  2, — 
The  quotient  is  .266  +  . 


103 


§166  ABSTRACT    NUMBERS. 


Explanation. — Since  the  dividend  can  be  extended  only 
by  annexing  naughts,  it  is  evident  that,  if  the  same  remain- 
der should  occur  twice  in  succession,  the  same  quotient 
figure  will  occur  and  will  give  rise  to  the  same  remainder 
again  ;  so  that  the  same  circuit  of  operations  will  occur  per- 
petually. In  such  cases  the  quotient  can  not  be  obtained 
exactly,  but  we  can  always  make  an  approximation  suffi- 
ciently near  for  any  practical  purpose. 

Rule. — Divide  as  in  ?'-7'o^/^  numbers,  and  point  off  as 
tnamj  fractional  figures  in  6-....  (quotient  as  the  numher  in  the 
dividend  exceeds  the  numher  in  the  divisor,  ijrefixtng  naughts 
when  necessary  to  make  up  the  numher. 

If  the  nmnher  of  fractioiial  figures  in  the  divisor  exceeds 
the  numher  in  the  dividend,  annex  to  the  dividend  as  many 
naughts  as  may  he  necessary  to  make  the  numher  in  the  divi- 
dend at  least  equal  to  the  number  in  the  divisor. 

Note. — When  the  division  can  not  be  exactly  performed, 
we  put  the  sign  +  at  the  right  of  the  quotient. 

Proof. — The  same  as  in  whole  numbers.  (§  53). 

Ex.  4.  Divide  1728  by  .12.' 

5.  Divide  1728  by  1.2. 

6.  Divide  172.8  by  12. 

7.  Divide  17.28  by  12. 

8.  Divide  13  by  245. 

9.  Divide  2.7  by  900. 

10.  Divide  189.75  by  .759.  Quot.  250. 

11.  Divide  84.099  by  .097.  Quot.  867. 

12.  Dividend  is  4435.2,  divisor  is  .84. 

13.  Dividend  is  .8928,    divisor  is  1.24.  Quot.  .72. 

14.  Dividend  is  7049.754,   divisor  is  8.7034.       Quot.  810. 

15.  Dividend  is  2.4416,   divisor  is  43.6. 

104 


CONTRACTED   DIVISION    OF   DECIMAL   FRACTIONS.    §168 


16.  Divisor  is  47,    divideDd  is  22.09. 

17.  Divisor  is  18.07,    dividend  i^  .12649. 

18.  Divisor  is  180.7,  dividend  is  .012649. 

19.  Divisor  is  .125,    dividend  is  2.25. 
L'O.  Divisor  is  18,    dividend  is  19. 


Quot.  .47. 
Quot.  .007. 

Quot.  18. 


CONTRACTION  IN  MULTIPLICATION. 


Ex.  1.  Multiply  23.25  by  10.     • 

§  167.    Model. — Remove  the 
23.25  X  10rr232.5         point   one   place   to  the   right. 

The  product  is  232.5. 

Explanation. — To  multiply  by  any  power  of  ten,  we 
simply  remove  the  units'  point  as  many  places  to  the  right 
as  there  are  naughts  in  the  multiplier,  annexing  naughts 
when  necessary.     See  §  155. 

Ex.  2.  Multiply  232.5  by  100. 

3.  Multiply  10.25  by  1000. 

4.  Multiply  246.25  by  100. 

5.  Multiply  875.275  by  10. 

6.  Multiply  96.0025  by  10000. 

7.  Multiply  .0025  by  1000. 

8.  Multiply  .0007  by  100000. 

9.  Multiply  .05  by  1000000. 
10.  Multiply  .0065  by  10000. 


Prod.  23250. 

Prod.  24625. 
Prod.  8752.75. 

Prod.  2.5. 
Prod.  70. 


Prod.  ()0' 


CONTIUOTION  IN  DIVISION. 

^■ 

Ex.  1.  Divide  23.25  by  10. 

§168.    Model. — liemove  the 
23.25 -f-lOmz 2.325         point  one  place  to  the  left.    The 

(juotient  is  2.325. 
105 


^169  ABSTRACT   NUMBERS. 


Explanation.^— To  divide  by  any  power  of  ten,  we  sim- 
ply remove  the  units'  point  as  many  places  to  the  left  as 
there  are  naughts  in  the  divisor,  prefixing  naughts  when 
necessary.     See  §155. 

Ex.  2.  Divide  2.325  by  100.  Quot.  .02325. 

3.  Divide  10.25  by  1000. 

4.  Divide  246.25  by  100.  Quot.  2.4625. 

5.  Divide  875.275  by  10.  Quot.  87.5275. 

6.  Divide  9G.0025  by  10000. 

7.  Divide  2500  by  1000.  Quot.  2.5. 

8.  Divide  7000  by  100000.  Quot.  .07. 

9.  Divide  .05  by  1000000. 

10.  Divide  .0065  bv  10000.  Quot.  .00000065. 


■<©>- 


RELATIONS  OF  COMMON  AND  DECIMAL 
FRACTIONS. 


§  169.  Every  decimal  fraction  may  be  expressed  in  the 
form  of  a  common  fraction  by  simply  removing  the  units' 
point,  writing  the  denominator  under  the  numerator,  and  re- 
ducing, if  necessary,  to  its  lowest  terms.  Thus,  .5=-!%-=:-^. 
Also,  .25=-^\\=l:. 

Ex.  1.  Reduce  .375  to  a  common  fraction.  Val.  |. 

2.  Reduce  .625  to  a  common  fraction.  Val.  ^. 

3.  Reduce  .1875  to  a  common  fraction. 

4.  Reduce  .3125  to  a  common  fraction.  Val.  -j^, 

5.  Reduce  .05  to  a  common  fraction.  Val.  -^^. 

6.  Reduce  .0015  to  a  common  fraction. 

7.  Reduce  00025  to  a  common  fraction;  Val.  ^o^. 

106 


COMMON    AND    DECIMAL   FRACTIONS.  .     §171 


8.  Reduco  .004375  to  a  common  fraction.  Val.  xoVo- 

9.  Reduce  ,08125  to  a  common  fraction. 

10.  Heduce  .0175  to  a  common  fraction,  Yal.  ^^-^^j. 

§170.  If  the  denominator  of  a  common  fraction  has  no 
other  prime  factor  than  2  or  5,  it  may  be  reduced  to  a  dec- 
imal form  by  multiplyiog  both  its  terms  by  such  a  number 
as  will  make  the  denominator  a  power  of  ten,  removing  the 
denominator,  and  putting  the  units'  point  at  its  proper  place 
in  the  numerator.  Thus,  multiplying  both  terms  of  -I  by 
25,  we  have  -jY^j  which  may  be  written,  .25. 

Ex.  11.  Heduce  |  to  a  decimal  fraction.  Yal.  .4. 

12.  Reduce  {:  to  a  decimal  fraction, 
18.  Reduce  f  to  a  decimal  fraction.  Val.  .625. 

14.  Reduce  ./^  to  a  decimal  fraction,  Val.  .35. 

15.  Reduce  ^}|  to  a  decimal  fraction. 

16.  Reduce  l^  to  a  decimal  fraction.  Val.  .475. 

17.  Reduce  ,.^^y  to  a  decimal  fraction.  Val.  .0375, 

18.  Reduce  '^-Jj  to  a  decimal  fraction. 

19.  Reduce  -^\  to  a  decimal  fraction.  Val.  .5625. 

20.  Reduce  -^^  to  a  decimal  fraction.  Val.  .09375. 

§  171.  If  the  denominator  of  a  common  fraction  has  nei- 
ther 2  nor  5  as  a  prime  factor,  it  cannot  be  reduced  to  a 
decimal  form.  We  can  make  an  approximation,  however, 
sufficiently  near  for  all  practical  purposes,  by  the  following 
plan.  Taking  the  example  of  last  section,  if  we  multiply 
both  terms  of  l  by  100,  we  have  i^,  and  then  dividing 
both  terms  by  4,  we  have  -^^^q,  that  is,  .25.  In  other  words, 
A  common  fraction  is  reduced  to  a  decimal  form  hi/  dividing 
its  numerator  hy  its  denominator,  (§  166).  This  is  the  gen- 
eral rule,  and  is  but  a  repetition  of  what  we  learned  in 
§  113.     But  let  us  attempt  to  apply  this  rule  to  the  frac- 

107 


§172  ABSTRACT    NUMBERS. 


tlon  |-.  I>ividing,  we  have  3  in  20,  6  times,  with  2  over ; 
again,  annexing  an  other  naught,  we  have  o  in  20,  6  times, 
with  2  over  ;  and  so  on,  evidently  forever.  Again,  reduce 
-i\  to  a  decimal  form.     Dividing,  we  have,  ll^n  20,  once, 

with  9  over ;  11  in  90,  8  times,  with  2 
11)2.0000  over;    11  in  20,  once,   with  9  over, 

.1818+         again  ;  and  11  in  90,  8  times,  with  2 
over,  again;  and  so,  evidently,  these 
two  quotient  figures  might  be  repeated  to  the  end  of  time. 
§  172.  Such  expressions  as  these  are  called ^wre  repctends, 
and  they  are  denoted  by  placing  a  dot  over  the  repeating 
figure  when  there  is  but  one,  or  by  placing  dots  over  the 
first  and  last  repeating  figures  when  there  are  several. 
Thus,  1^.6;  xV-.iS;  fff=.275. 
Ex.  21.  Reduce  ^  to  a  repetend. 

22.  Reduce  f  to  a  repetend.  Yal.  .285714. 

23.  Reduce  y\-  to  a  repetend.  Yal.  .2*7. 

24.  Reduce  -^-^  to  a  repetend. 

25.  Reduce  xV  ^o  ^  repetend.  Val.  .2941176470588235. 
§  173.  If  the  denominator  of  a  common  fraction  has  ei- 
ther 2  or  5  or  both,  and  other  prime  factors,  the  quotient 
of  its  numerator  by  its  denominator  will  be  partly  a  deci- 
mal fraction  and  partly  a  repetend. 

Thuo',  |=:.8333  +  ,  or  .83.     Also,  ^^j=A1666  +  ,  or  .416. 
Also,  -,\-=.2083  ;  and  -^^=.104<16. 

These  expressions  are  called  Tnixed  rcpetends. 

Ex.  26.  Reduce  -^^  to  a  mixed  repetend.  Val.  .585. 

27.  Reduce  ^  to  a  mixed  repetend. 

28.  Reduce  -^^  to  a  mixed  repetend.  yal.'.2i4285t. 

29.  Reduce  -^-^  to  a  mixed  repetend.  Val.  .46. 

80.  Reduce  |J-  to  a  mixed  repetend. 

108 


COMMON    AND    DECIMAL   J-RACTIOJSS.  §176 


§174.  To  reduce  a  jJiif'c^  rrpetcnd  to  a  common  fraction, 
toe  remove  the  units'  point,  vjrite  for  denomhiator  as  m'jny 
nines  as  there  are  repeating  figures,  and  reduce  the  result  to 
lis  lowest  terms. 

Again,  ,,V=-Oi.  -^=.05,  U=AO,  ^  =  .25,  f^-.50: 
Also, -,J^:=  .001,  WV=.010,  ^=.075,  ^^i]=:.275,  &c. 

§  175.  From  these  facts  we  learn  that  a  pure  rcpeteiid  is 
read  by  pronouncing  after  its  numerator  the  ordinal  of  the 
number  formed  of  as  many  nines  as  there  are  figures  in  the 
repetend.     Thus,  .1=1-,  .5^'7=|-^,  &c. 

Ex.  31.  Reduce  .27  to  a  common  fraction.  ^. 

82.  Reduce  .72  to  a  common  fraction.  .  -=& 


IT* 


33.  Reduce  .36  to  a  common  fraction. 

34.  Reduce  .185  to  a  common  fraction.  -^. 

35.  Reduce  .270  to  a  common  fraction.  -^j-, 

36.  Reduce  .7^2  to  a  common  fraction. 

37.  Reduce  .801  to  a  common  fraction.  -^^-^^-^ 

38.  Reduce  .9001  to  a  common  fraction.  uf-p* 

39.  Reduce  .8877  to  a  common  fraction. 

40.  Reduce  .9765  to  a  common  fraction.  i-i-fi- 
§  176.  A  mixed  repetend  is  a  complex  fraction,  having 

for  its  denominator  some  power  of  ten,  and  for  its  numera- 
tor a  mixed  number:  the  fractional  part  of  the  mixed  num- 
ber having  for  its  denominator  a  saries  of  nines. 

83  41 «  '2083- 

Thus,.85is--;.416is-^;.2083is^». 

To  reduce  a  mixed  repetend  to  a  simple  common  fraction, 

we  must  first  reduce  the  numerator  to  an  improper  fraction. 

This  makes  it  necessary  to  multiply  the  integral  part  by  9 

or  by  a  series  of  nines  ;  and  this  multiplication  can  be  most 

109 


§176  ABSTRACT    NU31BERS. 


readily  accomplished  by  §60.      Take  tLe  second  of  tbe 
above  examples,  for  instauce.     Annexing  one  naught  to  41, 

and  subtracting  41  from 

410     369     ^^100=^"^^     *^^  result,  we  have  369 

_il     __^     ~0~  '         ~"900     as  the  product  of  the  in- 

369     375  tegral  partby  the  denom- 

9  inator.     To  this  product 

adding  the  numerator  6, 
we  have  375  as  the  numerator  of  the  improper  fraction. 
Dividing  2.I-A  by  100,  we  find  -|-^,  which  should  then  be  re- 
duced to  its  lowest  terms. 

This  result  could  be  more  easily  obtained  by  f^uhtracting 
the  decimal ])art  from,  the  whole  repctend  for  the  numerator , 
and  hij  taking  for  the  denominator  as  many  nines  as  there 
are  repeating'  fgiircs,  followed  hy  as  many  naughts  as.  there 
are  decimal  figures. 

Thus,  .83r=i^=:f ;  .2083=(2083-208)=i§^^=:^ 
Ex.  41.  Pteduca  .123  to  a  common  fraction.  t^- 

42.  Keduce  .50t5  to  a  common  fraction. 

43.  Reduce  .779t  to  a  common  fraction.  -^-|-|. 

44.  Reduce  .1*76  to  a  common  fraction.  -^^. 

45.  Keduce  .4554  to  a  common  fraction. 


PROMISCUOUS  PROBLEMS. 


1.  What  is  the  sum  of  247  millionths,  26  ten-thousandths, 
163  hundred-thousandths,  3  thousandths,  and  19  hun- 
dredths ?  Ans.  .197477. 

2.  What  is  the  difi'erence  between  19  units  and  19  mil- 
lionths? Ans.  18.999981. 

110 


PROMISCUOUS   PROBLEMS.  §176 


3.  What  is  the  product  of  273  thousandths  and  11*7  ten- 
thousandths? 

4.  What  is  the  quotient  of  17  ten-thousandths  by  16  hun- 
dredths ?  Ans.  .10625. 

5.  What  is  the  Bum  of  the  product  of  5  tenths  and  5  hun- 
dredths, and  the  quotient  of  5  tenths  bj  5  hundredths  ? 

Ans.  10.025. 
G,   What  is  the  difference  between  the  sum  of  C  hundredths 
and  6  units,  and  the  product  of  6  hundredths  and  6 
units? 

7.  What  is  the  product  of  the  sum  of  12  thousandths  and 
84  hundredths,  and  their  difference?       Ans.  .115456. 

8.  What  is  the  quotient  of  the  product  of  506  thousandths 
and  78  hundredths  by  their  sum?        Ans.  .306905  +  . 

9.  Add   27   hundredths,  538  thousandths,   C4   ten-thou- 
sandths, and  9768  ni'llionths. 

10.  Subtract  the  product  of  39  hundredths  and  54  thou- 
sandths from  their  sum.  E-em.  .42294. 

11.  Multiply  the  quotient  of  36  hundredths  by  45  ten- 
thousandths  by  their  difference.  Prod.  28.44. 

12.  Divide  the  sum  of  497  thousandths  and  608   ten-thou- 

sandihs  by  their  difference. 

13.  What  number  is  that  to  which  if  13  hundredths,  13 
thousandths,  13  ten-thousandths,  and  13  millionthsbe 
added,  the  sum  will  be  13  units?         Ans.  12.855687. 

14.  What  number  is  that  from  which  if  11  hundredths,  12 
thousandths,  13  ten-thousandths,  and  14  hundred-thou- 
sandths be  subtracted,  the  remainder  will  be  15  mil- 
lionths?  Ans.  123455. 

15.  What  number  is  that  by  which  if  79  thousandths  be 
multiplied,  the  product  will  be  54115  billionths  ? 

Ill 


^176  ABSTRACT    NUMBERS. 


IG.   What  number  is  that  bj  whiich  if  6375   millionths  be 
divided,  the  quotient  will  be  5  thousandths  ? 

Ans.  1.275. 

17.  The  subtrahend  is  25  ten-thousandths,  the  minuend  is 
2  tenths;  what  is  the  remainder  ?  Ans.  .1975. 

18.  The  subtrahend  is  25  thousandths,  the  remainder  is  2 
hundredths  ;  what  is  the  minuend  ? 

19.  The  remainder   is   13   millionths,  the  minuend  is  13 
thousandths;"  what  is  the  subtrahend  ?     Ans.  .012987. 

20.  The  multiplicand  is  75  thousandths;  the  multiplier   is 
25  ten-thousandths  ;  what  is  the  product  ? 

Ans.  .0001875. 

21.  The  multiplier  is  18  thousandths,  the  product  is   369 
millionths;  what  is  the  multiplicand  ? 

22.  The  product  is  1482  ten-millionths,  the  multiplicand  is 
95  hundredths;  what  is  the  multiplier? 

Ans.  .000156. 

23.  The  divisor  is  19  huitdredths,  the  quotient  is  21  thou- 
sandths ;  what  is  the  dividend  ?  Ans.  .00399. 

21.  The  dividend  is  65  and  12  hundredths,  the   divisor  is 
17  and  6  tenths ;  what  is  the  quotient  ? 

25.  The  quotient  is  14  hundredths,  the   dividend   is   322 
thousandths  ;  what  is  the  divisor  t  Ans.  2.3. 

26.  What  are  the  prime  factors  of  3500  ] 

Ans.  2,  2,  5,  5,  5,  and  7. 

27.  What  are  the  prime  factors  of  756  ? 

28.  What  different  prime  numbers  will  exactly  divide  700  '^ 

Ans.  2,  5,  and  7. 

29.  What  different  prime  numbers  will  exactly  divide  850  ? 

Ans.  2,  5,  and  17. 

30.  What  is  the  least  common  multiple  of  7,  8, 10,  and  14? 

112 


PROMISCUOUS  PROBLEMS.  §176 


31.  What  is  the  smallest  number  that  may  be  exactly   di- 
vided by  either  9,  10,  12,  or  15  ?  Ans.  180. 

32.  What  is  the  smallest  number  that  may  be  exactly   di- 

vided by  either  21,  36,  48,  or  72  ?  Ans.  141. 

33.  What  is  the  greatest  common  measure  of  45,  54,  and 

108? 

34.  What  is  the  largest  number  that   will   exactly  divide 
either  75,  100,  or  150  ?  Ans.  25. 

35.  What  is  the  largest  number  that  will   exactly  .divide 
either  96,  192,  or  240  ?  Ans.  48. 

36.  What  is  the  sum  of  1-,  l,  -^,  |,  and  3%? 

37.  What  is  the  difference  between  f  and  |^?  Ans.  ^\. 

38.  What  is  the  product  of  |  and  j-^- 1  Ans.  |-|. 

39.  What  is  the  quotient  of  ^  divided  by  -^? 

7i 

40.  What  is  the  value  of  -^  ?  Ans.  |^. 

15i 

41.  What  is  the  value  of  f  of  -f-  of  7^-  ?  Ans.  If^. 

42.  What  is  the  value  of  .^25  +  . 025 +  .715 +  .225  ? 

43.  What  is  the  value  of  .0237-.002375  ?     Ans.  .021325. 

44.  What  is  the  value  of  .027  x  .0027  ?         Ans.  .0000729. 

45.  What  is  the  value  of  .0144-^3.6  ? 

46.  What  is  the  sum  of  -^,-i\,  yV?  and  ^  1  Ans.  l-i-|. 

47.  What  is  the  difference  between  -^\  and  -^  ? 

48.  What  is  the  sum  of  216  thousandths,  37  hundredths, 

15  ten-thousandths,  and  10  units?  Ans.  10.5875. 

49.  What  is  the  difference  between   206  ten-thousandths, 

and  27  millionths  ? 

50.  What  is  the  value  of  .211  +  3.07  +  29.6  +  .0735  ? 

Ans.  32.9545. 

51.  What  is  the  value  of  .6501  x  .736089  ? 

52.  What  is  the  value  of  .4396^9.3  ? 

n  113 


§177 


CONCRETE   NUMBERS. 


CONCRETE  NUMBERS. 


§  177.    The  relations  of  the  concrete  numbers  in  most 
common  use  are  set  forth  in  the  following 

I.    United  States  Money. 


1 

mill. 

(m.) 

—  1 

—  lO 

of  a 

cent; 

10  mills 

=  •      1  cent, 

(ct.) 

—  1 

10 

<i    a 

dime ; 

10  cents 

=          1  dime. 

(d.) 

—  1 

—  10 

a    ic 

dollar 

10  dimes 

=z         1  dollar, 

m 

—  1 

—  1  o 

U      il 

eagle ; 

10  dollars 

=          1 

eagle, 

(E.) 

m. 

ct. 

d. 

$ 

E. 

II   II   II   II 

rH  O  O  O 
rH  O  O 

1 

1  o 
1 

10 
100 

= 

1 

10  0 

1 

10 

1 

10 

^^          1000 

1 

—  1  oo 

—  1 

10 

=      1 

= 

1 

1  oooo 
1 

1000 

1 

100 

1 

1  0 

10000     = 

1000 

= 

100 

=      10 

— 

1. 

The  denominations  dime  and  eagle  are  very  little  used  in 
calculation.     In  stead  of  14E.  5$,  7d,  5ct.,  we  usually  write 

$145.75. 


II.    £n^lisli  Currency;    or.  Sterling  Mouej 

1  farthing,  (qr.)  - 

4  farthings      =:     1  penny,  (d.)  = 

12  pence            =     1  shilling,  (s.)  - 

20  shillings       =     1  pound,  {£). 

114 


i  of  a  penny; 
i-V  "  "  shilling; 
^  "  "  pound ; 


RELATIONS. 

§17' 

qr. 

d. 

s. 

£ 

1 

= 

1 

4.- 

—      1 

48 

*  = 

9  60 

4 



1 

1 

, 

1 

— 

12 

~^ — 

2  40 

48 

z=: 

VA 

:rr      1 

= 

1 
20 

960 

— 

240 

=:           20 

— 

1 

The  pound  sterling  is  represented  by  a  gold  coin,  called 
a  sovereign,  valued  at  $4.84,  U.  S.  currency. 

Farthings  are  usually  written  as  fractions  of  a  penny. 


III.    FreificU  Currency. 

1  centime,  (cent.)  =      J-  of  a  decimej 

10  centimes       =     1  decime,     (dec.)  =      /^  '^  "  franc ; 
10  decimes        =     1  franc,         (fr.)- 

cent.  dec.  fr. 

^  10  —  100 

10  =  1  =  iV 

100  =  10  =  1 

Accounts  are  kept  in  francs  and  centimes. 

The  franc  is  valued  at  18ct.  6m.,  U.  S.  currency. 


--«yp~ 


IV.    Troy  IRTeiglit. 

USED  FOR  WEIGHING  GOLD,  SILVER,  JEWELS,  <fec. 

1  grain,  (gr.)  =  ^l  of  a  pennyweight  ^ 
24  grains  =  1  pennyweight,  (dwt.)=  -^  of  an  ounce.; 
20  penny  weights^  1  ounce,  (oz.)=  -^^  of  a  pound  .; 
12  ounces     =  1  pound,      (lb.) 

gr.       dwt.       oz.        lb. 

1   =     -1-   =   -1-   =    1 

24  480      5760 

24   —     1    _   _i_   —    1 

^^     ^  20      240 

480     ::=:       20     =      1      =      JL. 

5760   =   240   =   12    =    1 
115 


§177 


CONCRETE    NUMBEHS. 


¥.    Apotliecaries'  W' 

elglit. 

USED  IN  MlJrjJVO  MEDICINES. 

1  grain,          (gr.) 

-  ^of 

a  scruple 

20  grains 

~     1  scruple, (so.)  or  9 

=    i    ^' 

"  dram  ; 

3  scruples 

=r     1  dram,  (dr.)  or  o 

6 

^'  ounce  • 

8  drams 

=     1  ounce,  (oz.)  or  5 

1        <6 

12 

"  pound  ^ 

12  ounces 

=     1  pound,  (lb.)  or  lb 

g^' 

sc.               dr. 

oz. 

lb. 

1 

^^             "2  0"             "6"0 

=      24        =       8       = 

1            — 

4  S  0           

1            — 

2  4              

,1               = 

1 

-20 
60 

480 

5  7  6  0 
1 

2  8  & 

1 

0  e 

1 
T2" 

:5760 

=    288        =96       = 

12        = 

1 

The  poun 

d  Apothecaries'  is  the  same  as  the  pound  Tro}^ 

¥1.    Avoirdupois  'Weaglat. 

USED  FOE,  WEIGHING  ALL  AllTICLES  EXCEPT  THOSE  MENTIONED 

ABOVE. 


•  16  drams 
16  ounces 
25  pounds  = 

4  quarters  = 

20  hundredweight: 

dr. 

1 

16 

256 

6400 

25600 

512000 


1  dram,     (dr..)=  -^  of  an  ounce ; 
=  1  ounce,    (oz.)=  J^-  of  a  pound ; 
=  1  pound,  (lb.)=  J-5  of  a  quarter  ; 
=  1  quarter,(qr,)=  ^of  ahundredweight' 
=  1  hundred weight,(cwt,)  =  -^V  of  a  ton; 


oz. 

1 

1  6 
1 

16 

400 

1600 

32000 


lion,    (T.). 

lb.    qr. 


1  . 

2  5  6' 
Ji._  . 
1  6   ■ 

1  : 

25  : 

100  : 

2000  : 

110 


1 

6  40  0 

1 

4  00 

1 

2  5 


cwt. 


2  5  6  00 

1 

16  0  0 

1 
100 


T. 


4 
80 


4   — 
1   = 

20   = 


5  12  000 

1 

3  2  0  00 

1 

2  0  0  0 

_1_ 

8  0 

1 
2  O 

1 


RELATIONS. 


§177 


144  pounds  Avoirdupois  =  175  pounds  Troy  or  Apothe- 
:  7000  gr.  Troy ;  1  oz.  Avoir.  =  437.5  gr. 


cariess 

1  lb.  Avoir. 
Troy. 

The  following  denominations  also  belong  here  : 


28  pounds 
112       " 
2240       " 
14       " 
21^  stone 
8    pigs 

50  pounds  of  salt 

56 

60 

56 
100 
]96 
200 


1  long  quarter ; 

1  long  hundredweight ; 

1  long  ton ; 

1  stone; 

1  pig; 

1  fother. 


"  corn 

'^  wheat 

''  butter 

"  salt  fish 

"  flour 

"  beef,  pork,  or  fish; 


1  bushel, 
1  bushel. 
1  bushel. 
1  firkin. 
1  quintal, 
1  barrel. 
1  barrel. 


Til.    Frencli  Wciglits. 

1  milligramme  =  -^-^  of  a  centigramme; 
10  milligrammes=l  centigramme 
10  centigrammes=l  decigramme   =  iV 
10  decigrammes  =1  gramme  ==  -^ 

10  grammes  =1  decagramme  =  -^-^ 

10  decagrammes  =1  hectogramme^=  -^ 
10  hectogrammes =1  kilogramme   =  y^^ 
100  kilogrammes  =1  quintal  =  y^^  "  "  millier; 

10  quintals  =1  millier,  or  1  ton  of  sea  water. 


yV  "  "  decigramme; 
^  ''  gramme  ; 
^  "  decagramme  ; 
^  "  hectogramme; 
^  "  kilogramme  ; 


i    a 


quin 


tal 


1  gramme  =  15.433  grains  Troy. 
117 


§177 


CONCRETE   NUMBERS. 


Till,    ftiong-  Measure 

^  or,  I^inear  Measure. 

USED  IN  MEASURING  LINES,  OR  DISTANCES. 

1  inch, 

(in.)  = 

=     yV    0^     ^     ^00*  J 

12  inches         — 

1  foot, 

(ft.)  = 

=   i    "  "  yard; 

3  feet             = 

1  yard, 

(yd.)  - 

-   ^\   "  "  rod  ; 

5i  yards 

1  rod, 

(rd.)  = 

=  ^^  "  "  furlong ; 

40  rods            — 

1  furlong 

,     (fur.)  = 

=    i   "  "  mile; 

8  furlongs     — 

1  mile, 

(mi.) 

in,               ft. 

yJ- 

rd. 

fur.          mi. 

1  —          '^    -- 

—              1 

1 

—      1      —       1 

-^                                 12 

3  6 

19  6 

7  9  20               63360 

12    =                 1      : 

] 

2 

. 1          1 

3 

3  3 

6  6  0                    5  2  8  0 

36  =           8    = 

=                    1 

2 

1          1 

1  1 

2  2  0                    1  7  60 

198  =r        16i= 

H 

=                 1 

1          1 

4  0                       ■3T0 

7920  —       660    : 

=       220 

=        40 

1                             i 

63360  =     5280    : 

=     1760 

=      320 

8—1 

The  following  denominations  are  sometimes  used  : 

3  barley 

corns  = 

1  inch  ; 

6  points 

— 

1  line  ; 

12  lines 

1  inch  ; 

4  inches 

= 

1  hand ; 

9     " 

— 

1  span  ; 

18     " 

:zjL 

1  cubit; 

21.9" 

1  sacred  cubit ; 

3  feet 

: — : 

1  pace; 

6  feet 

z= 

1  fathom ; 

1 

69i  miles 

1                      

1  degree  < 

of  latitude. 

IX.    Swrveyor's  I<oiig  Measure. 

7.92  inches 

1  link. 

00- 

2V  of  a  rod  ; 

25  links          = 

1  rod. 

(rd.)  = 

i   "  "  chain  ; 

4  rods           = 

1  chain, 

(ch.)  = 

-jV"  "furlong; 

10  chains        = 

1  furlong, 

(fur.)  - 

i    «  "  mile  ; 

8  furlongs    = 

1  mile, 

(mi.) 

118 

RELATIONS. 

§177 

in. 

7.92  — 

198  — 

792  = 

7920  = 

63360  = 

1. 

1    — 

25  — 

100  = 
1000  = 
8000  = 

rd.         ch. 

"2  5"              TcTo 

1  _  i  ^ 

4—1 

40—10     — 

320    —  80     — 

fur.            mi. 

1      —       1 

1000                   SOOO 

1      —       1 

40                         3  2  0 

1           1 

10                      »o 

1      =      i 
8—1 

X.    Square  Measure. 

USED  FOR  MEASURING  SURFACES   OF  LAND,  PAINTING,  PLASTERING. 

PAVING,    &c. 

1  squareinchj  (sq.in.)=y^-^ofasquarefoot; 

144  square  inches=l      "      foot,  (sq.ft. )=  I   '^  "     "    yard; 
9      «     feet      =1       ''    yard,(sq.yd.)=j-^y" '^  perch; 
30-i    «    yards  =1  perch,  (P.)    =  ^V  "  "  ^^^^  ; 

40  perches         =1  rood,  (R.)   =  |-  "  "  acre  ; 

4  roods  =1  acre,  (A.)  =^^0"  "square mile 

640  acres  =1  square  mile,  (sq.mi.). 

sq.in.sq.ft.sq.yd.     P.  R.  A.  sq.mi. 

^ ^~~  1  4 -i ~~ Ta ITe""  3  8  2  o ¥ "~~  1  5  6  8 "i~6  o"  —  "e "aT 2~6  4 0"       To  1  4  4  8  9  eoU 
144   —    1    —    1    —        4_    —    \ —  1 —  1- 

^^^     -^     V     10«9      10890  43560  '>.  7  k  7  s 

l9Qfi    —     q     _    1     _        t__    —        1        —    !„ 

39204   =    272:1    =   30^   z=   1   =   -,V  = 

1568160   =   10890   =   1210  =   40   =.: 

6272640   =  43560   =:  4840   =.    160  :.=   4   =    .   _    ^-^^ 

4014489600  =  27878400  =z  3097600  =  102400=2560=640=1 


XI.    Cubic  Measure. 

USED  FOR  MEASURING  THE  CONTENTS  OF  SOLIDS. 

1  cubic  inch,  (cu.in.)=yyi^-g-of  acubicft.:; 
1728  cubic  inches=l     "      foot,  (cu.ft.)  =   ^V  *'  "    "  J^-'^ 
27     ''     feet     =1     ''      yard,(cu.yd.). 

cu.  in.  cu.  ft.  cu.  yd. 

1  —  _3_.  —  L 

■^  1728  4  6050 

1728         =z  1  =  ^\ 

46656         =27  =  1 

119 


§177 


CONCRETE    NUMBERS. 


Also,  40  cubic  feet  of  round  timber  =  1  ton  ; 
50     "  "    ''  hewn  timber    =  1  ton  ; 

42     "  "    '•  shipping  =  1  ton ; 

16     "  ''    "  wood  =  1  cord  foot; 

128     "  "    ^'  wood  =  1  cord. 

Also,  231  cubic  inches=l  gallon  Liquid  or  Wine  Measure; 
268|-  "         "      =1       '^      Dry  Measure ; 
282     «         "      =1      "      x\le  Measure,  (out  of  use.) 
537-?-  "  "       =1  ponk; 

2150i  "         "      =1  Lu.liel. 


XII.    £iiq3£ld  Measure  5    or,  Wisae  Measiia'e. 

USED  IN  MEASURING  LIQUIDS  ;    AS,   MOLASSES,    SPIRITS. 
WINE,   WATER,   &c. 

1  gill,  (gi.)  =  I:  of  a  pint ; 

4  gills  =   1  pint,  (pt.)  —  1-  "  "  quart ; 

2  pints  =   1  quart,  (qt.)  =  -^  "  "  gallon  ; 

4  quarts  =z   1  gallon,  (gal.)  =  -^-3  "  "  barrel ; 

31-|- gallons  =   1  barrel,  (bbl.)  =:  -}  "  "  hogshead; 

2  barrels  —   1  hogshead(hhd.)  ==  ^  "  " 

2  hogsheads  =   1  pipe,  (pi.)  =  ^ 


pipe 
«    "  tunt 


2  pipes 

r=   1  tun, 

(tun). 

gi.          pt. 

qt. 

gal.      bbl.      hhd. 

pi.        tun. 

1   -       i 
4  =       1 

—           Y    — 

Z   126  Z 

1     —     1     —     1     — 

32       lOOb           3016 

s    ^^  "2T5'"2  ^^  "sinr 

4.       126    2  52 

•*-       0  3                   G  3 

3U-=    1    =    i    = 

1     —     1 

4  0  3  2          a  0  6  i 
1        1 

8  =       2 

32  =       8 

1008  =   252 

1  0  0  s       2  0  1  g: 
1     —     1 

504    100 tt- 

_1     —     1 

12  6    2  5  2 

2016  ==  504 

=  252  = 

63^=    2    =    i    = 

1         1 

2         4 

4032  :=1008 

=  504  = 

126  =    4    =    2    = 

1        =        i 

8064  =2016 

=  1008  = 

252  =    8    =    4    = 

2    =    1 

Also, 

42  gallons 

=   1  tierce ; 

2  tierces 

=   1  puncheon. 
120 

RELATIONS.  §177 


XIII.    Ale  Measaire. 

FORMERLY  USED  FOil  MEASURING  MALT  LIQUORS  AND    ?IILK,   WHTCli 

NOW.  HOWEVER,  ARE  GENERALLY  MEASURED  BY 

LIQUID  MEASURE. 

1  pint,  (pt.)   =    ^  of  a  quart ; 

2  pints         =   1  quart,  (qt.)   =   -}  "  "  gallon; 

4  quarts       =   1  gallon,  (gal.)   =  ^fo  "  "  barrel; 

:>6  gallons     =:   1  barrel,  (bbl.)   -   |  "  "  hogshead  ; 
li  barrels  =:   1  hogshead,  (hhd.) 

Also,  9  gallons  =  1  firkin, 

2  firkins    =  1  kilderkin. 


(qt.) 

(gal.) 
(pk.) 

-  1 

i 

1. 

1 

ii 

*'  gallon  ; 
"  peek ; 
"  bushel; 

,      (bu.) 

gal. 

pk. 

bu. 

1          — 

1 

1  6 

1 

—         1 

XIV.    itry  Measure. 

USED  FOR  MEASURING  GRAIN,  FRUITS,  VEGETABLES,  SALT,  Ic. 

1  pint,  (pt.)   =:   \-  of  a  quart ; 

2  pints       =   1  quart, 
4  quarts     —   1  gallon, 
2  gallons  —   1  peck, 
4  pecks      =    1  bushel, 
pt.  qt. 

1       -        i       = 

8  =r.  4  ^  i 

16:=.Sz::.2rrrl^A 

Also,  5  bushels  zzi   1  barrel,  of  corn  ; 
8  bushels  r:::   1  quarter ; 
36  bushels  —   1  chaldron. 
In  the  Confederate  States,  corn  is  usually  bought  and 
sold  by  the  barrel.     A  barrel  of  corn  should  contain  280 
pounds. 

121 


y,  3  2 

1  —  1 

a'  —         V 


§177 


CONCRETE   NUMBERS. 


V ^ — 

XV. 

Time. 

1  second. 

(sec.) 

1 
GO 

of 

a 

minute  ; 

60  seconds 

—   1  minute, 

(min.) 

T 
6  0 

an 

L  hour ; 

60  minutes 

=  1  hour, 

(hr.) 

1 
•J.  4 

a 

day; 

24  hours 

z=z   1  day, 

(da.) 

4. 
146  1 

a 

year  ; 

365^^  days 

=  1  year, 

(yr.) 

1 
1  0 

a 

decade; 

10  years 

=   1  decade. 

(dec.) 

1 
10 

ii 

century ; 

10  decades 

=  1  century, 

(cent.) 

sec.  min.  hr. 

da.     yr. 

dec. 

cent. 

I—  1  _  1  _ 

_  1  1 

1 

0  0"^ 

_ 

1 

•^   6  0   3  e  00 

S64;06    3155760O    3155760 

"yi  5  5  7  6  OOOO 

60  =  1  =  -i-. 

1          

1   

1 



1 

1      14  40  "     525960 

5  2  5  0  6  00 

52596000 

3600=  60  : 

==  1  =  ^v  - 

1 

1 

8  7  6  6  0 

—       1 

S  7  6  6 

S  7  6  6  00 

86400  =  1440  =  24  =  1 

146  1 

2 

1 

7  30  5 

3  6  5  2  5 

31557600  = 

525960  =  8766  =  365^  = 

1  ^ 

= 

1   1 

10       100 

315576000  = 

=  5259600=  87660  =  3652^  = 

=  10  = 

=  1  =  tV 

3155760000  =  52596000  =  876600  =  36525  =  100  =  10  =  1 

x\lso,  7  days  =   1  week,  (wk.)  ; 

30  or  31  days    =   1  month,  (mo.)  ; 
12  months    =   1  year. 

According  to  the  table,  3651  days  make  a  year.  To  ob- 
viate the  difficulty  arising  from  the  fraction,  we  reckon 
three  years  of  385  days  each,  and  one  of  866  days.  This 
long  year  is  called  leap  year.  The  leap  years  are  those 
whose  numbers  are  exactly  divisible  by  4 ;  except  that  the 
centennial  years  are  not  leap  years  unless  their  numbers 
are  exactly  divisible  by  400.  Thus,  1860  and  1848  were 
leap  years  ;  but  1900  will  not  be  leap  year,  because  it  is 
not  divisible  by  400. 

The  year  is  also  divided  into  four  seasons;  Spring,  Sum- 
mer, Autumn,  and  Winter.     These  consist  of  the  following 

months : 

122 


RELATIONS, 


§17' 


r    3.  March, 
Spring,    <    4.  April, 
I    5.  May, 

6.  June, 

7.  July, 

8.  Aucust, 


Summer, 


(Mar.)  lias  31  clays. 

(Apr.)  "  30  " 

(May)  "  31 

(Jun.)  "  30 

(Jul.)  "  31  '' 

.^...,       (Aug.)  "  31  " 

(    9.  September,(Sept.)  "  30  " 

Autumn,  MO.  October,      (Oct.)  "  31  '^ 

(  11.  November,(Nov.)  "  30  " 

r  12.  December,  (Dec.)  "  31  " 

Winter,  <     1.  January,     (Jan.)  '^  31  '^ 

{    2.  February,  (Feb.)  "  28  "  leap  year,  29. 

In  most  business  transactions  30  days  are  considered  a 
month. 


XVI.    Circular  Measure. 

USED  IN  SURTEYING,  GEOGRAPHY,  AND  ASTRONOMY. 

1  second,  ('' or  sec.)=g^^  of  a  minute; 
60  seconds  =1  minute,  ('or  min.)^:-^^  "  "  degree; 
60  minutes  =1  degree,  (°  or  deg.)=  ^^j  "  "•  sign  ; 
30  degrees  =1  sign,  (S.)=y\-  "  "  circumference; 

12  signs        =1  circiimference,(C.) 

fr 

1       = 

60     =  1 

3600     =  60 

108000     =        1800 

1^96000     =      21600 

Also,  60  degrees  =:   1  sextant     =-1-  of  a  circumference  ; 
And,  90       "         =1  quadrant  =a.  «  «  " 

123 


6  O 


o 

S. 

c. 

1 

= 

1 

= 

1 

36  OO 

10  8  000 

12  9  6000 

1 

= 

1 
1  S  00 

= 

1 

GO 

2  16  0  0 

1 

= 

1 

30 

= 

1 
360 

30 

= 

1 

z=z 

I 
1  2 

360 

=z 

12 

:;:;::: 

1 

§178 


CONCRETE    NUME"£:RS. 


24  sheets 
20  quires 
2  reams 
5  bundles 


XYII.    Faper. 
1  sheet,  (sh.) 

1  quire,         (q^-) 
1  ream,  (rm.) 

1  bundle,    (bdie.) 
1  bale. 


^13:  of  a  quire  ; 
2V  **  "  ream; 
i   "  "  bundle  ; 
i   "  "  bale: 


-•♦^ 


1  unit,  =  tV  of  a  dozen ; 

12  units     =z   1  dozen,  (doz.)   =  ^V  "  "  gi'oss; 

12  dozen    =   1  gross,  (gr.)     —   -,V  "  "  g^^^*  S^°^^  5 

12  gross     :=   1  great  gross. 

Also,  20  units  =    1  score. 


OPERATIONS   ON  CONCRETE  NUMBERS. 


The  numerical  processes  are  the  same  for  concrete  num- 
bers as  for  abstract.  In  this  place^  therefore,  we  are  to  dis- 
cuss only  the  denominations  of  the  several  results. 


ADDITION  OF  CONCRETE  NUMBERS. 


§178.  Dissimilar  numbers  can  not  be  added  together. 
Thus,  3  dollars  and  5  cents  malre  neither  8  dollars  nor  8 
cents. 

§  179.  The  sum  of  several  similar  numbers  is  similar  to 
the  numbers  added.  Thus,  3  dollars  and  5  dollars  make  S 
dollars  ;    3  cents  and  5  cents  make  8  cents. 

Ex.  1.  Add  $1075,  $2157,  $3779,  and  $4209, 

124 


ADDITION'.  !^i79 


2.  Add  ^47,  £bd,  £29,  nnd  £.63.  Sum,  £192. 

".  Add  27fr.,  3Gfr.,  297fr.,  and  365fr. 

4.  Add  291b., "STlb.,  491b.,  and  581b.  Sum,  1731b. 

5.  Add  45sc.,  2Si?c.,  143sc.,  and  2S7sc.  Sum,  5l.)3se. 
G.  Add  iOOcwt.,  205cwt.,  177cwt,,  and  329c wt. 

7.  Add  2479  grammes,  147 grammes,  and  986  grummeis. 

Sum,  3GI2graui. 

8.  Add  276yd.,  299jd.,  4G9yd.,  and  357yd. 

Sum,  140]  yd. 

9.  Add  79mi.,  227mi.,  37mi.,  and  475mi. 

10.  Add  306cu.  ft.,  279cu.  ft.,  and  520cu.  ft. 

Sum,  1105cu.ft. 

11.  Add  575A.,  209A.,  105A.,  and  258A.        Sum,  1147A. 

12.  Add  27gal.,  72ga].,  298gal.,  and  143gal. 

13.  Add  15bbl.,  28bbl.,  19bbl.,  247bbl.,  and  SGbbl. 

Sum,  395bbl. 

14.  Add  47bu.,  475bu.,  407bu.,  and  4750bii. 

Sum,  5679bu. 

15.  Add  27da.,  38da.,  52cla.,  and  93da. 

16.  Add  12°,  26°,  37°,  and  45°.  Sum,  120°. 

17.  Add  lOrui.,  14rm.,  7rm.,  and  22rm.  Sum,  53rm. 

18.  Add  6doz.,  27doz.,  14doz.,  and  97doz. 

19.  Add  12^-lb.,  33Ub.,  37ilb.,  and  83-^lb.      Sum,  166pb. 

20.  Add  3|mi.,  I6|mi.,  18:^ mi.,  62irai.,  and  42|mi. 

Sum,  143i{:mi. 

21.  Add  19^qt.,  20^qt.,  7^.qt.,  and  28^qt. 

22.  Add  3.251ir.,  6.5hr.,  .275hr.,  and  700.075hr. 

Sum,  710.1hr. 

23.  Add  47.5pt.,  57.75pt.,  .375pt.,  and  .0625pt. 

Sum,  105.6875pt, 

24.  Add  9.73pk.,  lO.Olpk.,  17.75pk.,  and  .1775pk. 

25.  Add  lO.llpk.,  7.369pk.,  and  1.002pk. 

125 


§180  CONCRETE   NUMBERS. 


SUBTRACTION  OF  CONCRETE  NUMBERS. 


§180.  Subtraction  can  not  be  performed  upon  dissimilar 
numbers.  Thus,  3  cents  from  5  dollars  leaves  neither  2 
cents  nor  2  dollars. 

§  181.  The  difference  of  two  similar  numbers  is  similar  to 
those  numbers.  Thus,  3  dollars  from  5  dollars  leaves  2 
dollars  ;  3  cents  from  5  cents  leaves  2  cents. 

Ex.  1.  From  £245  take  .£196.  Rem.  £49. 

2.  From  25cwt.  take  6c wt.  Kem.  19cwt. 

3.  From  793rd.  take  546rd. 

4.  From  17246sq.  ft.  take  8472sq.  ft.         Rem.  8774sq.  ft. 

5.  From  635cu.  yd.  take  473cu.  yd.  Rem.  162cu.  yd. 
(5.  From  47  decigrammes  take  29  decigrammes. 

7.  From  479bhd.  take  3981ihd.  Rem.  81hhd. 

8.  From  272pt.  take  199pt.  Rem.  73pt. 

9.  From  365da.  take  175da. 

10.  From  360°  take  275°  Rem.  85°. 

11.  From  27fs.  take  19is.  Rem.  8is. 

12.  From  $75^  take  $59i. 

13.  From  $19.75  take  $  .99.  Rem.  $18.76. 

14.  From  270^fr.  take  197|fr.  Rem.  72ifr. 

15.  From  77in.  take  17.75in. 

16.  From  3706sq.  yd.  take  897isq.yd.     Rem.  2808^sq.yd. 

17.  From  ^  of  246cu.  ft.  take  -\  of  317cu.  ft. 

18.  From  525iqt.  take  252^qt. 

19.  From  27bu.  take  I7.25bu.  Rem.  9.75bu. 

20.  From  SS^cu.  in.  take  31icu.  in.  Rem.  2y\cu.  in. 

21.  From  725dwt.  take  339.17dwt. 

22.  From  .2468d.  take  .08642d.  Rem.  .16038d. 

23.  From  .1751b.  take  .0171b.  Rem.  .1581b. 

126 


MULTIPLICATION. 


§183 


MULTIPLICATION  OF   CONCRETE  NUMBERS. 


§  182.  Every  muUipUer  must  be  an  abstract  number. — 
Thus,  if  we  wish  to  find  the  cost  of  o  yards  at  25  cents  a 
yard,  it  is  evidently  absurd  to  say,  "3  yards  times  25 
cents,"  or,  ''  25  cents  multiplied  by  3  yards."  "We  multi- 
ply 25  cents  by  3,  because  3  yards  cost  3  times  the  price  of 
1  yard,  that  is,  3  times  25  cents. 

§  183.  The  product  is  always  similar  to  the  multiplicand. 
Thus,  3  times  25  cents  are  evidently  75  cents ;  6  x  7  ab- 
stract units=:i42  abstract  units  ;  4  x  $10=$40  ;  5  x  6-yards 
=30  yards. 

Ex.  1.  Multiply  $3179  by  27. 

2.  Multiply  2764bu.  by  4G. 

3.  Multiply  3S5da.  by  19. 

4.  Multiply  347oz.  by  83. 

5.  Multiply  2047cwt.  by  109. 

6.  Multiply  347fr.  by  201. 

7.  Multiply  467A.  by  5297. 

8.  Multiply  6386pi.  by  578. 

9.  Multiply  7475pk.  by  689. 

10.  Multiply  £69  by  4234. 

11.  Multiply  224  by  4759. 

12.  Multiply  8564wk.  by  790. 

13.  Multiply  9563  by  801. 

14.  Multiply  10742doz.  by  912. 

15.  Multiply  20s.  by  16750. 

16.  Multiply  5}yd.  by  746. 

17.  Multiply  16.5ft.  by  165. 

18.  Multiply  30.25sq.  yd.  by  3.025. 

19.  Multiply  7.92in.  by  198. 

127 


Prod.  $85833. 
Prod.  127144bu. 

Prod.  28801OZ. 
Prod.  223123cwt. 

Prod.  2473699A. 
Prod.  3691108pi. 

Prod.  .£292146. 
Prod.  1066016. 

Prod.  7659963. 
Prod,  9769704doz. 

Prod.  4103yd. 
Prod.  2722.5ft. 

Prod.  1568.16in. 


i 


^184  CONCRETE   NUxMBERS. 


20.  Multiply  31igal.  by  1008.  Prod.  31752gal. 

21.  Multiply  SGo^da.  by  365^. 

22.  Multiply  $29.75  by  29.75.  Prod.  $885.0625. 

23.  Multiply  $100,375  by  37.5.  Prod.  $3764.0625. 

24.  Multiply  279.5  by  27.95. 


DIVISION  OF  CONCLIETE  xXUMBERS. 


{M84.  Division  ia  the  reverse  of  multiplication.  In 
multiplication,  the  two  factors  are  given,  to  find  tbe  prod- 
uct;  in  division,  the  product  and  one  of  its  factors  are 
given,  to  find  the  other  factor.  The  dividend  corresponds 
to  the  product;  the  divisor  may  correspond  to  either  the 
multiplicand  or  the  multiplier,  and  the  quotient  corre- 
sponds to  the  other. 
'  Thus,  6  X  25gal.=150gal. 

Conversely,  150gal.-^6=:25gal. 
Or,        150gal.-^25gal.=G. 

§  185.  Either  the  divisor  or  the  quotient  must  be  ab- 
stract, and  the  other  must  be  similar  to  the  dividend. 

In  other  words,  if  the  dividend  and  the  divisor  are  simi- 
lar, the  quotient  is  abstract :  if  the  divisor  is  abstract,  the 
(juotient  is  similar  to  the  dividend. 

The  remainder  is  always  similar  to  the  dividend.    (§46). 

Ex.  1.  Dividend=:45ct.,  divisor=^-3.  Quot.  15ct. 

2.  I)ividend==:$750,  divisor=$25.  Quot,  30. 

3.  Dividend=::=1000bu.,  divisor=40. 

4.  Dividend- 2451b.,  divisor=5.  Quot.  49^b. 

5.  Dividend=3003,  divisor=ll.  Quot.  273. 

6.  Dividendz=]728cu.  in.,  dmsor=48cu.  in. 

7.  Dividend=:7007yd.,  divisor=13yd.  Quot.  539. 

128 


REDUCTION.  §188 


8.  Divisor=l7mi.,  dividend=2S9mi.  Quot.  17. 

9.  Divisor=25,    dividends  1175gi. 

10.  Divisor=:27,    dividend=:2971ir.  Quot.  llbr. 

11.  Divisor=109,   dividend  =:2398qt.  Quot.  22qt. 

12.  Divisors 245cu.  ft.,   dividend  =  5880cu.  ft. 

13.  Divide  642780  dozen  by  36  dozen.  Quot.  17855. 

14.  Divide  79008oz.  by  96.  Quot.  823o2. 

15.  Divide  847665qr.  by  345qr. 

16.  Divide  3475cwt.  by  296.  Quot.  11.739  +  cwt. 

17.  Divide  1001s.  by  27s.  Quot.  37.074. 

18.  Divide  SS^^ft.  by  172ft. 

19.  Divide  372.25sq.  yd.  by  250sq.  yd.  Quot.  1.489. 

20.  Divide  iA.  by  13 1-.  Quot.  .03tA. 

21.  Divide  243flb.  by  19f. 

22.  Divide  799.6T.  by  87.'5T.  Quot.  9.104  + • 

23.  Divide  34.1.5  grammes  by  19.25  grammes. 

24.  Divide  177pt.  by  771.  Quot.  .2295  +  pt. 


KEDUCTION  OF  CONCRETE  NUMBERS. 


§  186.  A  compound  number  may  be  reduced  to  a  simple 
one,  or  a  simple  concrete  number  to  a-compound  one  by  tbe 
application  of  the  following  rule  according  to  tbe  circum- 
stances of  the  case. 

§187.  Rule. — Find  from  the  proper  table  the  value  of 
one  of  the  given  units  in  terms  of  the  required  denomination  ; 
and  multiply  this  value  by  the  number  of  the  given  units. 

Ex.  1.  Reduce  6  gallons  to  pints. 

§188.  Model.     6gal.=r6x4qt.  =  24qt. 

24qt.  =  24x2pt.=48pt. 

Hence,  6gal.=48pt. 
I  129 


§189  CONCRETE   NUMBERS. 


Explanation. —Since  4qt.=:lgal.,6gal.j  tliat  is,  6  times 
lgal.=6  times  4(it.  And  since  2pt.  =  lqt.,  24qt.,  or  24 
times  lqt.=24  times  2pt. 

Observe  that  in  each  instance  the  product  is  similar  to 
the  multiplicand.  (§183.) 

Otherwise^  6gal.  =  6  x  8pt.=:48pt. 
Ex.  2.  Reduce  6gal,  3qt.  Ipt.  to  pt. 
§189.  Model.     6gal.  =  6x4qt.r=24qt. 

24  +  3^:27     27qt.r=27  x  2pt.  =  54pt. 

54  +  1=55     Hence,  6gal.  3qt.  lpt.  =  55pt. 

Explanation. — After  reducing  the  6gal.  to  qt.,  the  giv- 
en 3qt.  may  be  added  to  the  result.  (§  179.)  And  after 
reducing  the  27qt.  to  pt.,  the  giv  s  Ipt.  may  be  added  to 
this  result. 

OtlieriDise,  6gal.=6  x  4qt.  =  24qt.=24  x  2pt.=:48pt. 

3  '^'  =  3x2^^=  6  " 

Hence,   6gal.  3qt.  lpt.=:55pt. 
Otherwise,  6gal.=6  x  8pt.r::48pt. 
3qt.  =3x2"  =  6" 

Ipt.  = 1_^ 

Hence,    6gal.  3qt.  lpt.  =  55pt. 
Evidently  the  final  result  is  not  affected  by  the  order  in 
which  the  several  reductions  are  performed. 
Ex.  3.  Eeduce  i^lb.  to  oz.,  dwt.,  &c. 
§  190.  Model.     TVlb.=yV  of  12oz.=l-^oz. 
^^oz.=|:  of  20dwt.  =  4dwt. 
Hence,  -Jylb.  =  loz.  4dwt. 
Explanation. — This  example  differs  from  the  first  only 
in  the  fact  that  here  each  multiplier  is  a  fraction. 

130 


REDUCTION.  §193 


Ex.  4.  Reduce  3795P.  to  A.,  R.,  &c. 
§191.  Model.  3795P.  =  3795x-J,tR.=H^R.  =  94R,  35P. 
94R.=:     94  X  1:A.=    V  A.  =  23A.    2R. 
Henoe,  3795P.  =  23A.  2R.  35P. 
Explanation. — This  example  differs  from  the  preceding 
only  in  the  fact  that  here  each  multiplicand  is  a  fraction. 
Ex.  5.  Reduce  6gal.  3qt.  Ipt.  3gi.  to  hhd. 
§192.  Model.     3gi.      =3     x  ipt.    =    a   pt, 

3^qt.     =3^-    X  igal.  =  U  gal. 
6||gal.  =  6|ixJ3hhd.=^^hhd. 
Explanation." — Here  both  factors  are  fractional. 

Ex.  6.  Reduce  30bu.  Ipk.  3qt.  Ipt.  to  pk. 
§  193.  Model.     30bu.  =r  30  x  4pk.  =  120pk. 
Ipk.  =     1  « 

3qt.=  3x  ipk.=       .375 

lpt.=:         t\V^'= •0625_ 

Hence,  30bu.  Ipk.  3qt.  Ipt."^21^43f5pk. 
Explanation. — This  example  is  but  a  combination  of 
two  of  the  preceding  ones,  and  seems  to   require  no   addi- 
tional explanation. 

Ex.  7.  In  $14,  how  many  mills  ?  Ans.  14000m. 

8.  In  jS15,  how  many  pence  ?  Ans.  3600d. 

9.  In  19fr.,  how  many  centimes  1 

10.  In  221b.,  Troy,  how  many  dwt.  ?  Ans.  5280dwt. 

11.  In  251b.,  Apothecaries',  how  many  scruples  ? 

Ans.  7200sc. 

12.  In  261b.,  Avoirdupois,  how  many  drams  ? 

13.  In  31  hectogrammes,  how  many  decigrammes  ? 

Ans.  SlOOOdec. 

14.  In  45  miles,  how  many  feet  ?  Ans.  237600ft. 

131 


S193 


CONCRETE   NUMBERS. 


Ans.  2048qt. 
Ans.  1728000sec. 

Ans.  8160sh. 
Ans.  720doz. 


15.  In  49fur.,  ho^  many  chains? 

16.  In  50 A.,  how  many  square  yards?     Ans.  242000sq.  yd. 

17.  In  65cu.  yd.,  how  many  cu.  in.  ?        Ans.  2566080cii.  in. 

18.  In  72gal.,  how  many  gi.  ? 

19.  In  64bu.,  how  many  qt.  ? 

20.  In  20da.,  how  many  sec.  1 

21.  In  29°j  how  many  seconds  ? 

22.  In  17rm.,  how  many  sheets  ? 

23.  In  5gr.  gross,  how  many  doz.? 

24.  In  M,  3s.  2d.,  how  many  qr.  ? 

25.  In  5fr.  7dec.  Scent.,  how  many  centimes  ? 

Ans.  578cent. 

26.  In  61b.  5oz.  3dwt.,  how  many  gr.  ?  Ans.  37032gr. 

27.  In  31b.  6oz.  5dr.  2sc.,  how  many  sc.  ? 

28.  In  28T.  lOcwt.  3qr.,  how  many  lb.  ?  Ans.  570751b. 

29.  In  1  millier,  5  quintals,  how  many  grammes  ? 

Ans.  1500000gr. 

30.  In  2rd.  3yd.  2ft.,  how  many  in.  ? 

31.  In  10  chains,  1  rod,  how  many  links  ?         Ans.  10251k. 

32.  In  4sq.  yd.  6sq.  ft.,  how  many  sq.  in.  ?     Ans.  6048sq.in. 

33.  In  lOcu.  ft.  400cu.  in.,  how  many  cu.  in.? 

34.  In  2  tuns,  Ipi.  Ihhd.,  how  many  gal.  ?         Ans.  693fral. 

35.  In  5bu.  2pk.  Igal.  3qt.,  how  many  pt.  ? 

36.  In  Icent.  6dec.  5yr.,  how  many  yr.  ? 

37.  In  2S.  25°  45',  how  many  seconds  ? 

38.  In  2rm.  lOqr.  12sh.,  how  many  sh.  ? 

39.  In  3gr.  4doz,,  how  many  units? 

40.  Reduce  -}£  to  s.  and  d. 
41.-  Reduce  ffr.  to  decimes. 

42.  Reduce  fib.  Troy,  to  oz.,  dwt.,  &c. 

43.  Reduce  ^  lb. Apothecaries',  to  oz.,  &c. 

Yal.  2oz.  3dr.  12ffr. 
132 


Ans.  366pt. 

Ans.  308700^ 
Ans.  1212sh. 

Val.  6s.  8d. 
Val.  6fdec. 


REDUCTION. 


§193 


44. 
45. 

46. 
47. 
48. 
49. 
50. 
51. 
52. 
53. 
54. 
55. 
56. 
57. 
58. 
59. 
60. 
61. 

62. 
63. 
64. 
65. 

66. 
67. 

68. 

69. 
70. 
71. 

72. 


Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
R.educe 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 

Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 

Reduce 
Reduce 
Reduce 
Reduce 


Val.  3cwt.  Iqr.  S^lb. 


Val.  22fP. 


^T.  to  cwt,,  qr.,  &c. 

^-mi.  to  fur.,  &c. 

f  A.  to  R.,  P.,  &c. 

;^rcu.  yd.  to  cu.  ft.,  &c.      Val.  4cu.  ft.  864cu.  in. 

f-gal.  to  qt.,  pt.j  and  gi. 

^jjhu.  to  pk.,  gal.,  &c. 

J  da.  to  lir.,  min.,  &c. 


Yal.2pk.  Igal.  2|qt. 
Val.  4lir.  48mm. 


Val.  2qr.  14.4sh. 
Val.  Idoz.  9.6un. 

Val.  lOoz.  8dwt.  8gr. 
Val.  lib.  4dr. 


.375°  to  min.  and  sec. 

.I3rra.  to  qr.  and  sh. 

.15gr.  to  doz.  and  units 

975qr.  to  £. 

oOOOgr.  to  lb. 

300sc.  to  lb.  Apothecaries' 

600000dr.  to  T. 

llOOOin.  to  mi.  Val.  Ifur.  15rd.  3yd.  2in. 

600001k.  to  mi.  Val.  7mi.  40ch 

4000000sq.  yd.  to  sq.  mi. 

60000cu.  in.  to  cu.yd. 

Val.  leu.  yd.  7cu.  ft.  1248cu.  in, 
lOOOOgi.  to  tuns.  Val.  1  tun,  60gal.  2qt 

lOOOpt.  to  lihd.,  A\q  Measure. 
250pt.  to  bu.  Val.  3bu.  3pk.  5qt, 

eOOOOOmin.  to  yr.  Val.  lyr.  51da.  16hr 

2000000"  to  circumferences. 
27000sh.  to  rra.  Val.  56rm.  5qr 

19000  units  to  gr.  gross. 

Val.  lOgr.  gr.  llgr.  lldoz.  4  units 
lid.  3qr.  to  £. 

9oz.  9dwt.  9gr.  to  lb.  Val.  .789 -fib 

6dr.  2sc.  15gr.  to  lb.  Val.  .07204  + lb, 

Iqr.  151b.  to  T. 


S19o  CONCRETE   NUMBERS. 


73.  Reduce  20rd.  Syd.  to  fur.  Val.  .52!>'7fur. 

74.  Reduce  2rd.  201k.  to  ch.  Val.  .7ch. 

75.  Reduce  IR.  lOP.  to  A. 

76.  Reduce  leu.  ft.  lOcu.  in.  to  cu.  yd. 

Val.  .03725 +  cu.  yd. 

77.  Reduce  Ipt.  Igi.  to  gal.  Val.  .15625gal. 

78.  Reduce  Ihhd.  Ibbl.  to  tuns. 

79.  Reduce  3pk.  Igal.  3qt.  to  bu.  Val.  .96875bu. 

80.  Reduce  lOhr.  15min.  BOsec.  to  da.  Val.  .4274  +  da. 

81.  Reduce  1*^  10'  30"  to  S. 

82.  Reduce  2qr.  12sli.  to  rm.  Val.  .125rm. 

83.  Reduce  Igr.  lOdoz.  10  units,  to  gr.  gross. 

Val.  .1585648igr.  gr. 

84.  Reduce  J12,  10s.  6d.  3qr.  to  s. 

85.  Reduce  iOlb.  9oz.  9dv/t.  9gr.  to  oz.     Val.  129.46875oz. 

86.  Reduce  31b.  5oz.  5dr.  Isc.  lOgr.  to  dr.       Val.  333. 5dr. 

87.  Reduce  IT.  lOcwt.  Iqr.  20ib.  to  cwt. 

88.  Reduce  Irai.  7fur.  20rd.  3yd.  to  rd.         Val.  620.o4rd. 

89.  Reduce  3ch.  2rd.  101k.  to  rd.  Val.  14.4rd. 

90.  Reduce  lOA.  3R.  20P.  to  R. 

91.  Reduce  2cu.  yd.  6cu.  ft.  75cu.  in.  to  cu.  ft. 

Val.  60.0434 +  CU.  ft. 
'92.  Reduce  lOgal.  Iqt.  Ipt.  3gi.  to  pt.  Val.  83.75pt. 

.93.  Reduce  2bu.  Ipk.  3qt.  to  pk. 

94.  Reduce  Ida.  Ihr.  Imin.  Isec.  to  min. 

Val.  1501.0l6min. 

95.  Reduce  1*=  10^  30''  to  minutes.  Val.  70^'. 

96.  Reduce  2rm.  3qr.  5sb.  to  qr. 

97.  Reduce  IT.  Icwt.  Iqr.  lib.  loz.  to  lb. 

Val.  2126.06251b. 

98.  Reduce  Isq.  yd.  Isq.  ft.  Isq.  in.  to  sq.  ft. 

Val.  10.00694sq.  ft. 
*-  134 


PROMISCUOUS  PROBLEMS.  §195 


PROMISCUOUS  PROBLEMS. 


1.  Bought  a  dress  for  $12,  a  cloak  for  $15,  a  bonnet  for 
$7,  and  a  pair  of  gloves  for  $1 :  what  did  they  all  cost  ? 

'  15  §  ^^^'  Model. — The  whole  cost  is  the  sum  of 
y  the  several  prices  :  hence,  add  $12,  $15,  $7,  and 
I  $1.  (§  179).  The  sum  is  $35  :  hence,  they  all 
cost  $35. 


2.  A  owns  $10475  in  real  estate,  $3850  in  slaves,  $4095 
in  good  notes,  and  $1415  in  cash ;  what  is  the  value  of 
his  whole  estate  ?  A.  $19835. 

« 

3.  Three  men  form  a  partnership  :  A  invests  $2445 ;  B, 
$2890  ;  G,  $1959  :  what  is  the  whole  investment  ?     . 

4.  A  miller  bought  from  one  man  147  bushels  of  wheat, 

from  an  other  98  bushels,  and  from  a  third  273  bush- 
els ;  how  mueh  wheat  did  he  buy  from  the  three  ? 

5.  A  farmer  raised  on  one  farm  415  bushels  of  wheat,  548" 
bushels  of  corn,  827  hundred  weight  of  hay ;  on  the 
other,  293  bushels  of  wheat,  487  bushels  of  corn,  2SQ 
hundred  weight  of  hay :  how  much  did  he  raise    on 
both  farms  ? 

A.  708bu.  wheat,  1035bu.  corn,  613cwt.  hay. 
().  Bought  a  farm  for  $2875,  and  sold  it  for  $3225 ;  what 
did  I  gain  t 

^""225  ^  ^^^'  ^^^^^'^- — ^^^^  g^^^^  '^^  *^®  difference 

9R7^  between  what  I  gave  and  what  I  received : 

-fiir  hence,   subtract  $2875  from  $3225.     (§181.) 

$350  The  difference  is  $350  :  hence,  I  gained  $350. 

7.  A  farmer  owning  725  acres,  sells  375  acres  ;  how  much 
land  has  he  remaining?  A.  350A. 

185 


§190  CONCRETE   NUMBERS. 


8.  A  man  divides  $3000  among  three  sons^  giving  A  ^985, 

and  B  $1235  :  how  much  does  he  give  C  ?       A.  $780. 

9.  Burnt  a  kiln  of  100000  bricks  ;  sold  at  different  times 

3475,  2800,  40150,  and  35000  ;   how  many  are  still 
unsold  ? 

10.  The  distance  from  Charlotte  to  Groldshoro',  via  High 

Point,  is  223mi.,  from  Charlotte  to  High  Point  is  79mi.; 
how  far  is  High  Point  fj;om  Groldshoro'  ?        A.  I4lmi. 

11.  What  cost  2471h.  of  hn^ -n.  at  19ct.  per  lb.? 

9A7     IQ  +  §186.  xiioDEL. — 2471b.  cost  247  times 

ooL^  the  cost  of  lib.:  hence,  multiply  19ct.,by 

-^  247.    (§183.)     The    product   is   4693ct.: 

4693ct.  hence,  the  bacon  cost  4693ct. 

12.  How  many  cents  are  in  25  dollars  ? 

13.  How  many  gallons  in  14hhcl.?  A.  882gal. 

14.  What  will  94bbl.  of  flour  cost  at  $8  per  bbl.?        $752. 

15.  How  many  pages  in  475  volumes  of  296  pages  each  ? 
IG.  A  father  divides  $5460  equally  among  his  4 sons;  what 

does  each  son  receive  ? 

§  197.  Model. — Each  son's  share  is  one 
4)$5460     fourth  of  the  whole  :  hence,  divide  $5460 
$1365     by  4.  (§185.)  The  quotient  is  $1365:  hence, 
each  son's  share  is  $1365. 

17.  If  755A.  of  land  cost  $12835,    what  will  one  acre  cost? 

A.  $17. 

18.  If  125  slaves  sell  for  $75125,    what  is  their  average 

value  ? 

19.  If  85  bales  of  cotton  weigh  386751b.,  what  does  each 

bale  weigh '/  A.  4551b. 

20.  On  475A.  of  land  I  raised  15675bu.  of  wheat ;   how 
much  per  acre  ?  A.  o3bu. 

21.  In  478241b.  flour,  how  many  bbl. 

136 


PROMISCUOUS  PROBLEMS.  §198 


^woci^ii,   1(^n^^.  ^198.  MoDEL.—As  1961b.  Blake  a 

478241b.  1961b.     i  ^  ,     xi  i         r  i,vi    •  i  x 

QP„         bbl.,  the  number  oi  bbl.  is  equal  to 

'^r'0  4       '^^^'         the  number  of  times  1961b.  are  con- 

f.r.r.  tained   in    478241b. :    hence,   divide 

^^^  478241b.  by  1961b.     (§185.)     The 

quotient  is  244  :  hence,  there  are  244bbl. 

22.  How  many  cu.  yd.  iu  13122cu.  ft.?  A.  486cu.  y^. 

23.  In  11823s.,  how  many  G.I  A.  563G-. 

24.  How  many  xVcres  can  be  bought  for  $5658  at  $23  per 

Acre  ? 

25.  If  a  vessel  make  376mi.  per  day,  how  long  will  she  be 
in  making  7144mi.?  A.  19da. 

26.  Find  the  sum  of  two  thousand  and  forty-seven,  three 
thousand  six  hundred  and  fifty,  sixty-three  thousand 
and  five,  ten  thousand  four  hundred  and  three,  and 
four  hundred  and  seven.  Sum,  79512. 

27.  Find  the  diftereuce  between  ten  thousand  and  forty- 

two,  and  eight  thousand  seven  hundred   and  ninety- 
nine. 

28.  What  is  the  product  of  seven  thousand  three  hundred 
and  seventy-five,  and  one  hundred  and  twenty-five? 

A.  921875. 

29.  IVhat  is  the  quotient  of  eight  thousand  six  hundred  and 

twenty-five,  by  one  hundred  and  twenty-five  ?     A.  69- 
80.  How  many  days  in  4wk.? 

31.  How  many  hours  in  28da.?  A.  672hr. 

32.  Hovv^  many  minutes  in  672hr.?  A,  40320min. 

33.  How  many  seconds  in  40320min.?" 

34.  The  minuend  is  91  thousand  8  hundred  and  75,  the  sub- 
trahend 8  thousand  9  hundi-ed  and  G9 ;  what  is  the 
remainder  ?  A.  82906. 

35.  The  subtrahend  is  4  thousand  2  hundred  and  96,  the 

137 


§198  CONCRETE   NUMBERS. 


remainder  6  thousand  2  hundred  and  84  :  what  is  the 
minuend  ?  *  A.  10580. 

26.  The  remainder  is  7  hundred  thousand  and  94,  the  minu- 
end 2  millions  3  thousand;  what  is  the  subtrahend? 

37.  How  many  hours  in  40320min.'?  A.  672hr 

38.  How  many  days  in  672hr.?  A.  2Sda. 

39.  How  many  weeks  in  28da.? 

40.  How  many  min.  in  2419200sec.?  A.  40320min. 

41.  The  multiplicand  is  37  millions  43  thousand  and  25, 
the  multiplier  8  thousand  and  64  ;  what  is  the  produef? 

A.  298714953600. 

42.  The  multiplicand  is  7  hundred  and  25,  the  product  593 
thousand  7  hundred  and  75  ;  what  is  the  multiplier  ? 

43.  The  multiplier  is  4  thousand  9  hundred  and  7,  the  prod- 

uct 42  millions  813  thousand  575  ;  what  is  the  mul- 
tiplicand ?  A.  8725. 

44.  What  cost  347yd.  of  rope  at  9ct.  per  foot  1     A.  $93.69. 

45.  How  many  qt.  in  7gal.  2qt.? 

46.  How  many  qt.  in  8gal.  Iqt.? 

47.  How  many  pt.  in  8gal,  Iqt.  Ipt.? 

48.  How  many  gi.  in  7gal.  3qt.  Ipt.  Sgi.l 

49.  The  dividend  is  11  millions  210  thousand  202,  the  di- 
visor 7  thousand  and  2  ;  what  is  the  quotient  ? 

A.  161. 

50.  The  divisor  is  8  thousand  and  4,  the  quotient  5  thou- 
sand and  90 ;  what  is  the  dividend  ?         A.  40740360. 

51.  The  quotient  is  1  million  2  thousand  and  3,  the  divi- 
dend 1  trillion  4  billions  10  millions  12  thousand  and 
9 ;  what  is  the  divisor  1 

52.  How  many  sq.  mi.  in  228S88P.? 

A.  2Rq.  mi.  150A.  211.  8P. 

53.  How  many  R.  in  1728P.?  A.  43R.  8P. 

138 


PROrvilSCUOUS   PROBLEMS.  §198 


54.  How  many  sq.  mi.  in  1895A.? 

55.  How  many  A,  in  1806P.?  A.  IIA.  IK.  6P. 

56.  What  is  the  sum  of  7  thousand,  83  thousand  and  40,  9 
hundred  and  70,  and  17  times  5  hundred  and  79  1 

A.  100853. 

57.  What  is  the  difference  between  the  product  of  85  and 
307,  and  the  quotient  of  999875  by  125  ? 

58.  How  many  lb.  in  7qr.?  A.  1751b. 

59.  How  mary  oz.  in  251b.  Avoir.?  A,  400oz. 

60.  The  distance  from  High  Point  to  Greensboro'  is  15mi., 
from  Greensboro'  to  Shops  22mi.,  from  Shops  to  Fta- 

4       leigh  53mi.;  how  far  is  it  from  High  Point  to  Raleigh, 
via  Greensboro'  and  Shops  ? 

61.  The  distance  from  Charlotte  to  High  Point  is  79mi., 
from  High  Point  to  Ilalcigh  95mi.,  from  Kaleigh  to 
Goldsboro'  49mi.;  hov/  far  is  it  from  Charlotte  to 
Goldsboro',  via  High  Point  and  Raleigh  ?     A.  223mi. 

62.  Bought  a  pair  of  horses  for  $375,  a  set  of  harness  for 

$55,  and  a  buggy  for  $187 ;  what  did  the  whole  cost  ? 

A.  $617. 

83.  Paid  $789  for  a  lot  of  tobacco,  and  sold  it  for  $910  ; 

gained  how  much  ? 

64.  How  many  units  in  14doz.  and  7  ?  A.  175. 

65.  How  many  units  in  3  score  and  10  ?  A.  70. 

66.  How  many  doz.  in  12  gross  ? 

67.  How  many  units  in  10  great  gross  ?  A.  17280. 

68.  Bought  3  stone  of  potatoes  at  2ct.  per  lb.;  what   did 

they  cost  ?  A.  84ct. 

69.  Bought  10001b.  of  fish  at  $9  per  quintal ;    what  did  I 

pay  ? 

70.  What  cost  6161b.  of  butter  at  $15  a  firkin  ?      A.  $175. 

71.  What  cost  247bbl.  of  flour  at  $5  per  bbl.?      A.  $1235. 

139 


§198  CONCRETE    NUMBERS. 


72.  How  far  will  a  traiu  of  cars  go  in  3  dajs,  at  16  miles 
per  hour  ? 

73.  Bought  16yd.  of   calico  at  15cfc.,  7yd.  of   gingham  at 

25ct.,  9yd.  of  flannel  at  GSct.,  and  25yd.  of  domestics  at 
lOct.;  paid  16bu.  of  corn  at  GSct.;  how  much  is  still 
due  ?  A.  $1.89. 

74.  If  a  hook  of  155  pages  has  29  lines  on  each  page,  and 

39  letters  in  each  line,  how  many  letters  are  in  the 
book  ?  A.  175305  letters. 

75.  I  deposited  in  bank  $10050  :  having  drawn  out  $15, 

$175,  $237,  $375,  $4165,  $394,  and  $3968,  how  much 
have  I  still  on  deposit  ?  ^ 

76.  The  Bible  contains  31173  verses  ;    how  many  verses 

must  I  read  each  day,  to  finish  it  in  one  year  1 

A.  85  verses  a  day,  and  148  verses  over. 

77.  How  many  sheets  of  paper  in  20  quires  ?  A.  480sh. 

78.  How  many  sheets  in  14  reams  ? 

79.  How  many  reams  in  180  quires  ?  A.  9rm. 

80.  How  many  quires  in  19  reams  ?  A.  380qr. 

81.  A  stock-dealer  bought  4'7  cows  at  $19,  29  horses  at 

$135,53  mules  at  $97,  and  155  sheep  at  $3:  he  received 
for  them  347  acres  of  land  at  $26^  and  $4125  in  money; 
how  much  did  he  gain  ? 

82.  What  will  574bbl.  of  pork  cost  at  $13  per  bbl.'? 

A.  $7462. 

83.  How  far  will  a  man  travel  in  6da.  at  29mi.  per  da.? 

A.  174mi. 

84.  A  planter  who  worked  57  hands,  raised  399  bales  of  cot- 

ton :  how  m.any  bales  did  he  raise  to  the  hand  ? 

85.  In  $45,  how  many  ct.?  A.  4500ct. 

86.  In  M,  5s.  6d.,  how  many  d.?  A.  1026d. 

140 


PROMISCUOUS   PROBLEMS.  ijlDb 


87.  In  240dwt.,  boyr  raaDj  oz.? 

88.  In  39sc.j  liow  many  dr.?  A.  I3dr. 

89.  In  3T.  3qr.  201b.  12oz.,  how  many  oz.?         A.  97532oz. 

90.  In  7920in.,  how  many  yd.? 

91.  In  4mi.,  how  many  ch.?  A.  320ch. 

92.  In  1568160sq.  in.,  how  many  sq.  yd.?        A.  1210sq.  yd. 

93.  In  4sq.  mi.,  how  many  A.? 

94.  In  4cu.yd.  12cn.ft.,  how  many  cu.in.?    A.  207360cu.in. 

95.  In  3025gi.,  how  many  hhd.?  '  A.  Ihhd.  31gal.  2qt.  Igi. 

96.  In  Sbu.j  how  many  pt.? 

97.  In  3da.  lOhr.  15mija.,  how  many  sec?       A.  296l00scc. 

98.  In  3S.  3°  3'  3'^,  how  many  seconds  ?  A.  334983''. 

99.  In  2gr.  gr,  3gr.  4doz.  and  5,  how  many  units  ? 

100.  In  6rm.  7qr.  8sh.,  how  many  sh.?  A.  3056sh. 

101.  In  3gal.  3qt.  8gi.,  how  many  qt,?  A.  15.375qt. 

102.  In  lObu'.  Ipk.  Igal.  Ipt.,  how  many  pk.? 

103.  In  ^6,  6s.  6d.  3qr.,  how  many  s.?  A.  126.5625s. 

104.  Add  4:1b. 5  -^-oz.,  -idwt.,  and  -Vgr.,  in  gr. 

Sum,  1688.2gr. 

105.  Add  3.5hr.5  7.75min.,  and  .15sec.,  in  min. 

106.  Add  iA.,  A.R.,  and  -^j^V.,  in  P.       .  Sum,  48.1P. 

107.  Add  .25cu.  yd.,  .375cu.  ft.,  and  .625cu.  in.,  in  cu.  ft. 

Sum,  7.12536 +  CU.  ft. 

108.  From  .9cwt.,  take  .251b.,  in  oz. 

109.  From  .751b.,  take  .5dwt.,  in  oz.  Rem.  8.975oz. 

110.  From  10.875s.,  take  9.15^.,  in  qr.  Rem.  485.4qr. 

111.  From  5.5da.,  take  5.5min.,  in  min. 

112.  Multiply  .75gal.  by  7.5,  in  pt.  Prod.  45pt. 
118.  Multiply  2.25A.  by  .125,  in  P.          *  Prod.  45P. 

114.  Divide  4.5mi.  by  5.4,  in  rd. 

115.  Divide  1.55s.  by  2.3,  in  d.  Quot,  8.08695d. 

141 


§199  COMPOUND   NUMBERS. 


OPERATIONS  ON  COMPOUND  NUMBERS. 


§  199.  The  operations  on  compound  numbers  are  analo- 
jxous  to  the  corresDondino;  ones  on  abstract  -iiumbers. 


ADDITION  OF  COMPOUND  NUMBERS. 


Ex.  1.  Add  together  4hhd.  25gal.  3qt.,  5hhd.  20gal.  2qt., 
7hhd.  17ga].  2qt. 

§  200.  MoDEL.~2  and  2  are  4, 
4hhd.  25gal.  oqt.  ^nd  3  are  7,  7qt.,  equal  to  Igal. 
5  ''  20  "  2  '^  3(^1^^^  gg.  (lo^jj  3  .  1  and  17  are  18, 
7  "      17  ''     2  '-      ^^^  20  are  38,  and  25  are  63,  63 


17  "        0  "     3  "      gal.,  equal  to  Ihhd.,  set  down  0 

1  and  7  are  8,  and  5  are  13,  and  4 
are  17,  i7hhd.     The  sum  is  17hhd.  3qt. 

Explanation. — In  simple  numbers  ten  units  of  any  de- 
nomination make  one  of  the  next  higher.,  In  compound 
numbers  this  uniformity  of  relation  does  not  exist.  Thus 
in  the  example  above,  4qt.  make  Igal.,  but  63gal.  make  1 
hhd.  With  this  exception,  the  explanation  in  §  22  will 
suffice  for  this  case. 

Ex.  2.  Add  ^10,  14s.  9d.  3qr.,  J^5, 16s.  6d.  2qr.,^7,10d. 
Iqr.,  .212,  9s.  9d.  3qr. 

3.  Add  ^4,  10s.  lid.,  £J,  8s.  9d.  3qr.,  ^8, 10d.,and  16s. 
3qr.  Sum,  =£20,  16s.  7d.  2qr. 

4.  Add  101b.  lOoz.  lOdwt.  lOgr.,  121b.  9oz.  6dwt.  3gr., 
91b.  lloz.  I3dwt.  15gr.,  and  24ib.  Boz.  15dwt.  20gr. 

Sum,  581b.  4oz.  6dwt. 

5.  Add  31b.  6oz.  9dwt.  12gr.,  61b.  8oz.  lOdwt.  12gr.,  81b. 
lloz.  14dwt.  17gr.,  and  141b.  lloz.  8dwt.  5gr. 

142 


SUBTRACTION.  §201 


6.  Add  101b.  9ok.7dr.2sc.l5gr.,10oz.6dr.  Isc.  lOgr:,  15 

lb.  lloz.  7dr.  2gc.  19gr.,  and  31b.  4oz.  5dr.  6gr. 

Sum,  3  lib.  loz.  3dr.  Isc.  lOgr. 

7.  Add  lOT.  lOcwt,.   lOlb.  lOoz.  lOdr.,  14T.   15cwt.  3qf. 

151b.  13oz.  15dr.,  and  25T.  7cwt.  Iqr.  201b.  8oz.  €dr. 
Sam,  SOT.  IScwt.  Iqr.  221b.  loz.  Idr. 

8.  Add  Ssq.mi.  300A.  2Fi..  25P.,  7sq.mi.  525A.  311.  lOR, 

19sq.mi.  285A.  3R.  19P.,  and  250A.  25P. 

Sum,  31sq.mi.  82A.  IR.  39P. 

9.  Add  19ou.yd.  19cu.ft.  19cu.in.,  25cu.yd.  25cu.ft.  250cu. 
in.,  and  lOOcu.jd.  IScu.ft.  1555cu.in. 

10.  Add  4bhd.  40gal.  2qt.  Ipt.  3gi.,  lOhbd.  lOgal.  Iqt.  Ipt. 

Igi.,  and  20bhd.  43gal.  3qt.  Ipt.  3gi. 

IBum,  35hhd.  32gal.  3gi. 

11.  Add  lObu.  Spk.  7qt.  Ipt.,  9bu.  2pk.  6qt.  Ipt.,  16bu. 

3pk.  Gqt.,  and  15bu.  Ipk.  5qt.  Ipt.  Sum,  53bu.  Iqt.  Ipt. 

12.  Add  30da.  ;LOhr.  30min.  SOsec,  15da.  15hr,  15min.  15 
sec,  and  lOda.  20hr.  45rain.  15s8c. 

13.  Add  25<^  15'  25'',  75°  24'  50",  and  15°  50'  45'^ 

Sum,  116°  Sr. 

14.  Add  2rm.  lOqr.  12sh.,  4rm.  15qr.  ISsh.,  and  3rm.  9qr. 
lOsh. 

15.  Add  2gr.  gross,  10  gross,  7doz.  5  units,  4gr.  gross,  8 
gross,  6doz.  7  units,  and  5  gross,  8doz.  6  units. 


SUBTRACTION  OF  COMPOUND  NUMBERS. 


Ex.  1.  From  o-ei7,  5s.  6d.  3qr.,  take  ^8,  10s.  9d.  2qr. 
o-i"      -     PA    ^  §201.  Model. — 2  from  3  leaves 

q'  io^;  o  a  Sir     1 ;  ^  ^^'01^  18  ^e^ves  9  ;  11  from  25 
. .^ilLj__l_     leaves  14 ;  9  from  17  leaves  8.  The 

8,  14"  9  "  1  «      remainder  is  ^8,  14s.  9d.  Iqr. 

143 


§201  COMPOUND   NUMBERS. 


Explanation. — As  9d.  can  not  "be  taken  from  6d.,  we  add 
Is.,  that  is  12d.,  to  the  minuend,  and  subtract  9d.  from  18d. 
We  then  add  Is.  to  the  subtrahend,  and  proceed.  See 
^§28,  30. 

Ex.  2.  From  501b.  6oz.  15dwt.  19gr.,  take  101b.  17dwt. 

Rem.  401b.  5oz.  ISdwt.  19gr. 

3.  From  151b.  logr.,  take  121b.  9oz.  lOdwt.  12gr. 

4.  From  lOT.  lOcwt.  lOoz.,  take  5T.  15cwt.  20ib.  12oz. 
lOdr.  Rem.  4T.  14cwt.  3qr.  41b.  13oz.  6dr. 

5.  From  6sq.mi.  2R.,  take  375A.  25P. 

Rem.  Ssq.mi.  265A.  IR.  15P. 

6.  From  250cu.yd.  20cu.ft.  875cu.in.j  take  79cu.yd.  25cu. 

ft.  G95cu.in. 

7.  From  15T.   15cwt.  3qr.   151b.,  take  lOT.  19cwt.  3qr. 
191b.  Rem.  4T.  15cwt.  Sqr.  211b. 

8.  From  lOhhd.  lOgal.  Iqt.  Ipt.,  take  9hhd.  33gal.  3qt. 

ogi.  Rem.  39gal.  2qt,  ]gi. 

9.  From  4  tuns,  Ipi.  Ihhd.  5gal.  2qt.  3gi.,  take  2  tuns,  60 
gal.  oqt.  3gi. 

10.  From  175bu.  Ipk.  3qt.  Ipt.,  take  54bu.  3pk.  2qt. 

.  Rem.  120bu.  2pk.  Iqt.  Ipt. 

11.  From  27bu.  2pk.  Ipt.,  take  ISbu.  5qt. 

Rem.  14bu.  Ipk.  oqt.  Ipt. 

12.  From  30da.  lOhr.  15min.,  take  17da.  15hr.  15sec. 

13.  From  180°,  take  74°  14'  45".  Rem.  105°  45'  15". 

14.  From  90°,  take  35°  41'  15".  Rem.  55°  18'  45". 

15.  From  100°  17'  30",  take  90°  25'  45". 

16.  From  22T.  Scwt.  2qr.  201b.,  take  12T.  18cwt.  221b. 

Rem.  9T.  lOcwt.  Iqr.  231b. 

17.  From  16hhd.  24gal.  3qt.  2pt.,  take  I4hhd.  37gal.  3qt. 

18.  From  236bu.  2pk.  5qt.  Ipt.,  take  17bu.  2pk.  7qt.  2pt. 

144 


MULTIPLICATION.  §202 


MULTIPLICATION  OF  COMPOUND  NUMBERS. 


Ex.  1.  Multiply  £11,  5s.  6d.  3qr.  by  15. 

i^l7,  5s.  6d  3qr.         §202.  Model.— 15  x  3=45.  45qr.= 

^^  lid.  Iqr.;    sefc  down  1:    15x6=90, 

259,  3  "  5  "  1  "      nnd  11  =  101.    101d.=8s.5d.;  set  down 
5:   15x5=75,  and  8  =  83.    88s.  =  ie4, 

OS. ;  set  down  3  :  15  x  17=255,  and  4=259.     The  product 

is  £2bd,  3s.  5d.  Iqr. 

Explanation.— See  §§  38,  200. 

Ex.  2.  Multiply  £bS,  10s.  9d.  2qr.  by  4. 

Prod.  ^214,  3s.  2d. 

3.  Multiply  13°  15'  45"  by  7. 

4.  Multiply  25°  30'  45"  by  10.  Prod.  255°  T  30". 

5.  Multiply  501b.  6oz.  15dwt.  19gr.  by  13. 

Prod.  6571b.  4oz.  5dwt.  7gr. 

6.  Multiply  121b.  9oz.  lOdwt.  12gr.  by  16. 

7.  Multiply  5T.  15cwt.  201b.  l2oz.  lOdr.  by  19. 

Prod.  109T.  8cwt.  3qr.  171b.  9oz.  14dr. 

8.  Multiply  2sq.  mi.  200A.  2R.  20P.  by  22. 

Prod.  50sq.  mi.  573A.  3R. 

9.  Multiply  3cu.  yd.  25cu.  ft.  750cu.  in.  by  25. 

Prod.  98cu.yd.  14c u.  ft.  1470cu.  in. 

10.  Multiply  9hlid.  33gal.  3qt.  3gi.  by  21. 

11.  Multiply  2  tuns,  60gal.  3qt.  3gi.  by  24. 

Prod.  53tijps,  Ipi.  Ihhd.  llgal.  Iqt. 

12.  Multiply  25bu.  3pk.  Iqt.  Ipt.  by  29. 

13.  Multiply  lObu.  Ipk.  4qt.  by  35.  Prod.  363bu.  4qt. 

14.  Multiply  lOda.  lOhr.  lOmin.  lOsec.  by  41. 

Prod.  lyr.  62da.  8hr.  56min.  50sec. 

15.  Multiply  17da.  15min.  15sec.  by  50. 

J  145 


§203  COMPOUND   NUMBERS. 


DIVISION  OF  COMPOUND  NUMBERS. 


Ex.  1.  Divide  15°  15'  50''  by  10. 

10)15°  15'  50"         §203.    Model.— 10   in   15,   once, 
1°  31'  35"     with  5  over,  set  down  1 ;    10  in  315, 
31  times,  with  5  over,  set  down  31 : 
10  in  350,  35  times.     The  quotient  is  1°  31'  35". 

Explanation. — 10  is  contained  once  in  10;  so  that  there 
are  5°  undivided.  These  5°  are  reduced  to  300',  and  added 
to  the  15',  making  315'.  In  like  manner,  the  5'  undivided 
are  reduced  to  300",  and  added  to  the  50",  making  350". 

Ex.  2.  Divide  £^0,  16s.  2d.  Iqr.  by  3. 

3.  Divide  £Q0,  Is.  5d.  by  4. 

4.  Divide  291b.  2oz.  2dwt.  2gr.  b;,  :». 

Quot.  51b.  lOoz.  lOgr. 

5.  Divide  2421b.  5oz.  lldwt.  16gr.  by  8. 

Quot.  301b.  3oz.  13dwt.  23gr. 

6.  Divide  4481b.  lOoz.  I4dr.  by  11. 

7.  Divide  32T.  2qr.  by  15.        Quot.  2T.  2cwt.  2qr.  201b. 

8.  Divide  52sq.  yd.  5sq.  ft.  128sq.  in.  by  20. 

Quot.  2sq.  yd.  5sq.  ft.  lOOsq.  in. 

9.  Divide  97cu.  yd.  22cu.  ft.  SOcu.  in.  by  26. 

10.  Divide  91gal.  Iqt.  Ipt.  by  34. 

Quot.  2gal.  2qt.  Ipt.  2gi. 

11.  Divide  79  tuns,  Iqt.  Igi.  by  45. 

Quot.  Ftun,  Ipi.  Ihhd.  Igal.  Iqt.  Ipt.  Igi. 

12.  Divide  600bu.  3pk.  6qt.  by  60. 

13.  Divide  8wk.  3da.  71ir.  43min.  20sec.  by  70. 

Quot.  20hr.  20min.  20sec. 

14.  Divide  1150°  31'  15"  by  75.  Quot.  15°  20'  25". 

15.  Divide  1°  41' 40"  by  100. 

146 


I'ROMISCUOUS  PROBLEMS.  §201 


J 6'.  Divide  57T.  lOcwt.  Iqr.  171b.  14oz.  by  9cwt.  Iqr.lTlb. 
lOoz. 

57T.  lOcwt.  Iqr.  171b.  14oz.  =  1855086oz. 
Ocwfc.  Iqr.  171b.  10oz.  =  15082oz. 
.  1855086oz.-^15082oz.  =  123. 

§  204.  MoDEL.—Recluce  the  dividend  to  oz.  (§  189).  Ke- 
duce  the  divisor  to  oz.  (§  189).  Divide  the  dividend  by  the 
divisor.  (§  185).     The  quotient  is  123. 

Ex.  17.  Divide  294lbii.  by  45bu.  3pk.  6qt.  Ipt. 

Quot.  G4. 

18.  Divide  97T.  llcwt.  3qr.  111b.  lOoz.  by  IT.  6cwt.  2qr. 

261b.  lOoz. 

19.  Divide  17bu.  Ipk.  6qt.  by  2bu.  Spk.  5qt.  Quot.  6. 

20.  Divide  51A.  IR.  IIP.  by  lA.  IP.  Quot.  51. 

21.  Divide  10  tuns,  2hhd.  ITgal.  2pt.  by  39gal.  6pt. 

22.  Divide  ^£27,  2s.  6d.  by  15s.  Gd.  Quot.  35, 


PROMISCUOUS  PROBLEMS. 


1.  What  is  the  least  common  multiple  of  15,  24,  and  27  ? 

2.  What  is  the  least  common  multiple  of  9,  25,  and  45  ? 

L.  C.  M.  225. 

3.  What  is  the  greatest  common  measure  of  505,  1111, 

and  3434? 

4.  What  is  the  greatest  common  measure  of  1015,  1260, 
and  1330?  G.  C.  M.  35. 

5.  What  are  the  prime  factors  of  6105 1 

6.  What  are  the  prime  factors  of  4060  ? 

7.  Divide  ^£113,  13s.  4d.  by  31.  Quot.  £S,  13s.  4d. 

8.  Divide  10  tuns,  Ipi.  17gal.  2pt.  by  67. 

9.  Divide  50T.  4cwt.  2qr.  I4lb.  by  23cwt.  3qr.  171b. 

147 


§204  PROMISCUOUS  PROBLEMS. 


10.  Divide  1572yd.  by  32yd.  3qr.  Quot.  48. 

11.  Multiply  25oz.  8dwt.  17gr.  by  100. 

Prod.  2111b.  lloz.  lOdwt.  20gr. 

12.  Multiply  21da.  181ir.  42niin.  by  75. 

13.  Subtract  40A.  3R.  25P.  from  79A.  15P. 

Rem.  38A.  30P. 

14.  Subtract  4  tuns,  Ipi.  llihd.  25gal.  3qt.  from  5  tuns,  Iqt. 

15.  Add  60mi.,  40mi.  7fur.  39rd.,  and  19mi.  Ird. 

16.  Add  13°  14'  15",  16°  17'  25",  25°  19'  47",  and  3°  15". 

17.  Divide  $1521808938  by  234.  Quot.  $6503457. 

18.  Divide  I42651ihd.  by  45hhd. 

19.  Multiply  4327bu.  by  102.  Prod.  44l354bu. 

20.  Multiply  47935gal.  by  275.  Prod,  13182l25gal. 

21.  Subtract  2598328fur.  from  3002575fur. 

22.  Subtract  187564329gi.  from  923465781gi.     ' 

Rem.  73590l452gi. 

23.  Add  2479A.,  3580A.,  1358A.,  and  9745A. 

24.  Add  ^613575,  £2Udb,  £9475,  and  £31525. 

25.  Divide  82960332  by  84.  Quot.  987265  ;  Rem.  72. 

26.  Divide  82071  by  99.  Quot.  829. 

27.  Multiply  24068  by  13579. 

28.  Multiply  1020908  by  8979091.    Prod.  9166825834628. 

29.  Subtract  3987456002  from  4567398745. 

30.  Subtract  246  +  357  +  1298  +  982  from  3120. 

31.  Add  20030405,  910285,  5821090,  and  9706845. 

Sum,  36468625. 

32.  Add  123,  1234,  12345,  123456,  1234567,  12345678, 

and  123456789.  Sum,  137174192. 

33.  A.  raised  125  bales  of  cotton,  517bu.  corn,  629bu.  wheat, 

and  119bu.  rye  ;  B.,  217  bales  of  cotton,  865bu.  corn, 

798bu.  wheat,  and  143bu.  rye ;  C,  94  bales  of  cotton, 

424bu.  corn,  517bu.  wheat,  and  77bu.  rye;   and  D., 

148 


PROMISCUOUS  PROBLEMS.  §204 

111  bales  of  cotton,  512bu.  corn,   558bu.  wheat,  and 

98bu.  rye.  How  much  of  each  article  did  they  all 
raise  ? 

84.  A  farmer  went  to  town  with  $100,  and  spent  $9  for 
molasses,  $13  for  sugar,  ^11  for  coffee,  $8  for  rice^ 
$17  for  dry  goods,  and  $25  for  leather.  Horv  much 
money  had  he  left  ?  Ans.  $17^ 

ii5.  Bought  47  acres  of  land  at  $19,  5  horses  at  $125,  10 
head  of  cattle  at  $21,  14  sheep  at  $3,  and  a  two-horse 
wagon  for  $65  ;  what  did  they  all  cost  ?     Ans.  $1835. 

36.  Sold  75  firkins  of  butter  for  $1350  ;  how  much  was 
that  a  firkin  ? 

37.  What  number  is  that,  to  which  if  245,  379,  124,  212, 
and  399  be  added,  the  sum  will  be  1525  ?      Ans.  166. 

38.  What  number  is  that,  from  which  if  the  sum  of  245, 
379,  124,  212,  and  399  be  subtracted,  the  remainder 
will  be  1525?  Ans.  2884. 

39.  AVhat  number  is  that,  by  which  if  twice  19  be  multi- 
plied, the  product  will  be  the  difference  between  4127 
and  2759  ? 

40.  What  number  is  that,  by  which  if  4235  bo  divided,  the 

quotient  will  be  77?  Ans.  55. 

41.  A  miller  has  5  bins,  one  of  which  holds  43bu.  3pk.  5qt.; 

the  second,  39bu.  Ipk.  3qt. ;  the  third,  45bu.  Iqt.  Ipt.; 
the  fourth,  53bu.  2pk. ;  the  fifth,  34bu.  3pk.  Ipt,  i 
what  is  their  united  capacity?     Ans.  216bu.  2pk,  2qt. 

42.  How  much  time  elapsed  between  Jan.  20th,  1833,  and 

May  25th,  1861  ? 

43.  Bought  4  lots  of  land,  containing  3B.  27P,  each  ;  how 

many  A.  did  I  buy  ?  Ans.  3A.  2B.  28P. 

44.  A  wine  merchant  has  269gal.  2gi.  of  wine  in  30  equal 

vessels;  how  much  wine  is  there  in  each  vessel  ? 

149 


P04 


PROMISCUOUS   PROBLEMS. 


45. 

46. 

47. 

48. 

49. 

50. 

51. 

52. 

53. 

54. 

55. 

50. 

57. 

58. 

59. 

60. 

61. 

62. 

63. 

64. 

65. 

66. 

67. 

68. 

69. 

70. 

71. 

72. 

73. 

74. 

75. 

76. 


Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

,Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

R.educe 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 


Val.  69119gr. 


£40,  19s.  lid.  3qr.  to  qr. 

111b.  lloz.  19dwt.  23gr.  to  gr. 

aib.  II5.  75.  29.  lOgr.  to  gr.       Val.  2B039gr, 

2T.  19cwt.  3qr.  24lb.  15oz.  15dr.  to  dr. 

4L.  2mi.  7fur.  35rd.  to  rd.  Val.  4795rd. 

12yd.  2ft.  lliu.  to  in.  Val.  467in, 

2sq.  mi.  600A.  3R.  35P.  to  P. 

25sq.  yd.  8sq.  ft.  lOOsq.  in.  to  sq.  in. 

21sq.  mi.  250A.  2R.  to  R.  Val.  54762R 

5cu.  in.  20cu.  ft.  1600cu.  in.  to  cu.  in. 


3  tuns,  Ipi.  llibd.  to  bhd. 
ohbd.  60gal.  3qt.  Ipt.  to  pt 
2gal.  Iqt.  Ipt.  3gi.  to  gi. 
5bu.  3pk.  7qt.  Ipt.  to  pt. 
30.  75yr.  300da.  to  da. 
4da.  lObr.  25min.  to  sec. 
25°  10'  J 5"  to  seconds. 
2rm.  15qr.  ]2sb.  to  sh. 

4  score  and  5  to  units. 
lOOOqr.  to  £. 
6000gr.  to  lb.  Troy. 
eOOOOOdr.  to  T. 
lOOOrd.  to  mi. 
2000sq.  in.  to  sq.  yd. 
200000sq.  rd.  to  sq.  mi. 
60000cu.  in.  to  cu.  yd. 
lOOgi.  to  gal. 
500qt.  to  bhd. 
lOOpt.  to  bu. 
4000sec.  to  hr. 
200br.  to  wk. 
4000"  to  degrees. 

150 


Val.  15bbd. 
Val.  1999pt, 

Val.  383pt. 
Val.  137175da. 

Val.  90615". 
Val.  1332sh. 

Val.  £1,  lOd. 
Val.  lib.  lOdwt. 


Val. 


om. 


Ifur. 


Val.  3gal.  Ipt. 

Val.  Ibu.  2pk.  2qt. 
Val.  Ibr.  6min.  40sec. 

Val.  1°  6'  40". 


PROMISCUOUS  PROBLEMS.  §204 


77.  Bought  8  firkins  of  butter  at  20ct.  per  lb.,  20qt.  mo- 
lasses at  $1  per  gal.,  3  stone  of  potatoes  at  3ct.  per  lb., 
and  9801b.  flour  at  $S  per  bbl.;  what  did  they  all  cost  ? 

78.  Bought  5doz.  Arithmetics  at  30ct.  apiece  ;    sold  them 

all  for  $27 :  how  much  apiece  did  I  gain  or  lose  ? 

79.  A  owes  B  for  5001b.  of  salt  fish   at  $8  a  quintal ;    B 

owes  A  for  8bbl.  flour  a^  $7  a  bbl.,  lObu.  corn  at  70ct. 
per  bu.,  and  5bu.  rye  at  80ct.  per  bu. :  how  does  their 
account  stand  ?  A  owes  B  $18. 

80.  From  500  subtract  the  sum  of  225, 120,  and  75  ;  divide 

the  remainder  by  the  diff"erence  between  1000  and 
960 ;  multiply  the  quotient  by  17 ;  and  add  16  to  the 
product.  Sum,  50. 

81.  Find  the  sum  of  the  product  of  88  and  11,  and  the  quo- 

tient of  88  by  8. 

82.  A  man  has  1184bu.  of  wheat  and  468bu.  of  corn,  which 
he  wishes  to  pack  in  equal  bags  as  large  as  possible. 
How  many  bushels  will  each  bag  hold  ;  and  how  many 
bags  will  be  required  ?  Ans.  4bu.,  and  413  bags. 

83.  What  is  the  value  of  85  +  77-64  +  6  x  19-132-^-4  ? 

84.  What  is  the  value  of  15498-^54  +  41  x  63-27  x  55  ? 

85.  Find  the  least  common  multiple  of  4,  44,  132,  and  792. 

86.  Find  the  greatest  common  measure  of  4,  44,    132,  and 

792.  G.  C.  M.  4. 

87.  Two  men  travel  in  the  same  direction  from  the  same 
place,  one  40mi.  a  day,  the  other  33mi.  a  day  ;  how  far 
apart  are  they  in  7  days  ? 

88.  Two  men  travel  in  contrary  directions  from  the   same 

place,  one  40mi.  per  day,  the  other  33mi.  per  day ; 
how  far  apart  are  they  in  7  days?  Ans.  511mi. 

89.  If  10  persons  use  a  barrel  of  flour  in  57  days,  how  long 
will  a  barrel  last  one  person  ?  Ans.  570da. 

151 


§204  PROMISCUOUS  PROBLEMS. 


90.  "What  ig  the  sum  of  3  numbers,  of  which  the  first  is  28, 
the  second  8  times  the  first,  and  the  third  one  seventh 
of  the  second  ? 

01.  The  difference  is  one  hundred  thousand  four  hundred 
and  seventy-six,  the  minuend  is  one  million  ;  what  is 
the  subtrahend  ?  Ans.  899524. 

92.  The  minuend  is  one  hundred  thousand,  the  subtrahend 
is  sixty-seven  thousand  seven  hundred  and  forty-four; 
what  is  the  remainder  ?  Ans.  32256. 

93.  The  subtrahend  is  f:.  ■>  i  Iiundred  thousand  and  forty- 
nine,  the  remainder  is  ninety-nine  thousand  two  hun- 
dred and  seventy-eight ;  what  is  the  minuend? 

94.  The  multiplicand  is  thirty-six  thousand  seven  hundred 

and  seven,  the  multiplier  is  eighty  thousand  and  one ; 
what  is  the  product?  Ans.  2936596707. 

95.  The  multiplier  is  eight  h-T^idred  and  four,  the  product 

is  sixty-one  thousand  nine  hundred  and  eight ;  what  is 
the  multiplicand  ?  Ans.  77. 

96.  The  product  is  eighteen  billions  two  hundred  and  twen- 
ty thousand,  the  multiplicnnd  is  two  thousand  two 
hundred  ;  what  is  the  multiplier? 

97.  The  divisor  is  one  hundred  and  twenty-five,  the  divi- 

dend is  nine  hundred  and  eighty-seven  thousand  six 

hundred  and  twenty-five;  what  is  the  quotient? 

Ans.  7901. 

98.  The  dividend  is  thirty-four  thousand  eight  hundred 
and  forty-eight,  the  quotient  is  one  hundred  and  thir- 
ty-two ;  wliat  is  the  divisor  ?  Ans.  264. 

99.  The  quotient  is  thirty  thousand  and  seventy,  the  divi- 
sor is  seven  hundred  and  eight ;  what  is  the  dividend? 

100.  What  cost  ISrm.  of  paper  at  $4,  3doz.  Arithmetics  at 
$6  a  dozen,  and  24  Algebras  at  812  a  dozen  ? 

152 


ALIQUOT   PARTS.  §20G 

ALIQUOT  PARTS. 


§  205.  An  aliquot  fraction  is  a  simple  fraction  whose  nu- 
merator is  1.     Thus,  ^V,  jf,  -},  To>  -^Vj  ^^®  aliquot  fractionj^!. 

An  aliquot  part  of  a  number  is  a  part  denoted  by  an  ali- 
(|uot  fraction.  Thus,  ?>  is  an  aliquot  part  of  12,  10s.  is  an 
aliquot  part  of  ^61,  20da.  is  an  aliquot  part  of  2mo. 

Ex.  1.  What  is  the  cost  of  5A.  3R.  25P.  of  land  at  $45. 

50  per  A.? 

2)  $45.50=1A.  ,  §206.  M0DEL-5A  cost 

^  -  5  times  as  much  as  lA. ; — 


0 


hence,  multiply  the  cost  of 
$227.50r:r5A.  YK  ^    5    (§183).     2R.  is 

^)     ""*"•/ ^   "tS"  ^  Slf-  ^^^  ^^^  *^^'  \k.\  hence,  di- 

2)     11.375=rlR.  S       '         vide  the  cost  of  lA.  by  2. 

4)       5.687=:20P.  ^  o^p       (§185).     IR.  is  one  half  of 

_J^421=r  5P.  j         •      2E.;  hence,  divide  the  cost 

$268,733  of  2R.  by  2.     20P.  is  one 

half  of  III. ;  hence,  divide 
the  cost  of  IR.  by  2.  5P.  is  odc  fourth  of  20P.;  hence,  di- 
vide the  cost  of  20P.  by  4.  Add  the  several  costs  together. 
The  sum  is  $268,733  :   hence,  5A.  3R.  25P.  cost  $268,733. 

Ex.  2.  What  is  the  cost  of  17T.  15cwt.  3qr.  101b.  of  iron 
at  $36  per  T.?  Ans.  $040.53. 

3.  What  is  the  cost  of  lObu.  3pk.  4qt.  Ipt.  of  grass  seed 
at  $8  per  bu.? 

4.  What  is  the  yield  of  45A.  3R.  15P.  of  wheat  land  at 

20bu.  per  A.?  Ans.  916.875bu. 

5.  What  is  the  value  of  21b.  9oz.  lOdwt.  6gr,  of  plate  at 

$15  per  lb.?  Anso'?4l.88. 

6.  What  is  the  value  of  5T.  lOcwt.  Iqr.  51b.  of  hay  at  $25 
per  T.? 

153 


§207  ALIQUOT    TARTS, 


7.  What  is  the  cost  of  15bu.  Ipk.  5qt.  of  dried  peaches  at 
80  per  bu.?  Ans.  692.4375. 

8.  What  is  the  cost  of  4gal.  Iqt.  Ipt.  of  wine  at  $4  per 
gal.?  Ans.  817.50. 

9.  What  is  the  co.st  of  37rd.   2^jd.  of  fencing  at  $3.50 
per  rd.? 

10.  What  is  the  cost  of  7yd.  2ft.  Gin.  of  cloth  at  $7.25  per 
yd.?  Ans.  s856.791  +  . 

1 1.  AVhat  is  the  cost  of  SOsq.yd.  4sq.ft.  72sq.iu.  of  painting 
at  $.75  per  sq.yd.?  '  Ans.  $22,875. 

12.  What  is  the  cost  of  lOcd.  SOcu.ft.  of  wood  at  $2.50 
per  cd.? 

13.  What  will  a  man  earn  in  9mo.  lOda.  at  $25  per  mo.? 

Ans.  $233.a3-J. 

14.  What  will  lOcd.  of  wood  cost  at  $2.62^  per  cd.? 

2)$10      =cost  of  lOcd.  at  $1.00    per  cd. 


$20 

— 

a 

a 

a 

2.00 

(( 

i( 

4)     5 

— 

a 

u 

a 

.50 

i( 

iC 

1. 

'2d= 

.12.^ 
2.02^ 

a 

$20. 

25= 

(( 

§  207.  Model,—  10  cords  at  $1  would  cost  $10.  The 
cost  at  $2  is  twice  the  cost  at  $1 ;  hence,  multiply  the  cost 
at  $1  by  2.  The  cost  at  50ct.  is  one  half  of  the  cost  at  $1; 
hence,  divide  the  cost  at  $1  by  2.  The  costatl2ict.  is  one 
fourth  of  the  cost  at  50ct.;  hence,  divide  the  cost  at  50ct. 
by  4.  Add  the  several  costs  together.  The  sum  is  $20.25: 
hence,  10  cords  at  $2.G2|  cost  $20.25. 

15.  What  is  the  cost  of  3G0bu.  of  wheat  at  $1.37-^-  per  bu.? 

16.  What  is  the  cost  of  15yd.  of  cloth  at  £1,  4s.  9d.  per 
yd.?  Ans.  ^18,  lis.  3d. 

154 


CONTRACTED    MULTIPLICATION   AND   DIVISION.        §209 


CONTRACTION  IN  MULTIPLICATION. 


Ex.  1.  Multiply  279  by  33^. 

q^97qnn         ^^^^'  ^-foDEL.— Annex  2  nauahts  to  the 
o;_wJuu     multiplicand  :— divide  the  result'by  o     The 
9300     product  is  9300. 

Explanation.— See  §63.  When  the  multiplier  is  an  ali- 
quot part  of  any  power  of  ten,  we  may  abridge  the  work  by 
multiplying  by  this  power  of  ten  (§56)  and  dividing  the  re- 
sult by  the  denominator  of  the  aliquot  fraction. 

Ex.  2.  Multiply  72  by  12i  Prod.  900. 

.3.  Multiply  72  by  16|-. 

4.  Multiply  77  by  14f .  Prod.  1100. 

6.  Multiply  as  by  li  Protl.  70. 

6.  Multiply  684  by  166§. 

7.  Multiply  273  by  333-}.  Prod.  91000. 


CONTRACTION  IN  DIVISION. 


Ex.  1.  Divide  2000  by  142^. 

2000         §209.  Model.— Multiply  the  dividend  by 

7     7  :— divide  the  product  by  1000.  (§  65.)     The 

14  000     quotient  is  14. 

Explanation.— See  §  68.  When  the  divisor  is  an  aliquot 
part  of  any  power  of  ten,  we  may  abridge  the  work  by  mul- 
tiplying the  dividend  by  the  denominator  of  the  aliquot 
fraction  and  dividing  the  product  by  the  power  of  ten.  {^65.) 

Ex.  2.  Divide  150  by  33^.  Quot.  4.5. 

3.  Divide  250  by  14|-. 

4.  Divide  1500  by  166|-.  Quot.  9. 

155 


^209  rROMiscuous  phoblems. 


5.  Divide  245  by  12 i.  gaot.  19.G. 

C.  Divide  1375  by  11^-. 

7.  Divide  2468  by  333^.  Quct.  7.404. 

PROMISCUOUS  PPtOBLEMS. 


1.  Find  246  +  2468  +  24680  +  20468  +  24068  +  24608. 

Sura,  96538. 

2.  Find  the  sum  of  one  half,  three  fourths,  two  fifths,  and 
one  sixth.  Sura,  1|^. 

3.  Add  315,  31.57,  3157,  3.157,  and  .3157. 

4.  A  farraer  raised  $357  worth  of  corn,  $475.50  worth  of 
wheat,  $123.75  worth  of  rye,  and  $446.37-^  worth  of 
other  products  ;  what  was  the  total  value  of  the  prod- 
ucts of  his  farm  ?  Ans.  $1402.62-v. 

5.  A's  gold  mine  yielded  -j^lb.  in  one  week,  B's  yielded  7 
oz.,  and  C's  125.5dwt.;  how  many  ounces  did  the  three 
mines  yield  ?  Ans.  19.275oz. 

6.  A  tobacco  planter  raised  from  one  field  7cwt.  Iqr.  20 

lb.,  from  an  other  lOcwt.  oqr.  101b.,  and  from  an  other 
15cwt.  151b.;  what  amount  did  ho  raise  in  all? 

7.  A  man  sold  14  horses  at  $125.75  a  head,  and  25  head 
of  cattle  at  $19.87^  a  head ;  what  did  he  receive  for 
them  all '/  "  Ans.  $2257.375. 

8.  Find  the  value  of  135-1357  +  13570-3571-3157. 

Yal.  5620. 

9.  What  is  the  difference  between  thirteen  fourteenths  and 
fourteen  fifteenths? 

10.  From  500.05  take  65.556.  Rem.  434.494. 

11.  A  dealer  sold  a  lot  of  bacon  for  $2425.10^,  which  cost 
him  $2177.4';^  ;  wliat  did  he  gain  by  the  trade  ? 

156 


PROMISCUOUS  PROBLEMS.  S200 


12.  Find  the  difference  in  K.  betvreeii  ^A,  and  27.5P. 
18.  Wasliiugton  was  born  Feb.  22,  1732,  and  died  Dec.  14, 
1799  ;  what  was  his  age?  Ans.  OTyr.  9mo.  22da. 

14.  A  gave  B  2501b.  of  beef  at  8ct.  for  o?Ah.  of  leather  at 

60ct.;  how  does  their  account  stand  '/ 

Ans.  B  owes  A  20ct. 

15.  What  is  the  product  of  7903  x  3907  ? 

16.  Multiply  three  eighths  by  two  and  two  thirds. 

Prod.   1. 

17.  Find  the  value  of  19.275  x  21.125.       Yal.  407.184375. 

18.  A  planter  who  works  47  hands  raises  to   each  hand  ten 

bales  of  cotton  averaging  4461b.;  how  much  does  his 
cotton  jaeld  him  at  9ct.  per  lb.? 

19.  A  farmer  owns  4  farms  containing  each  47A.  25P.;  what 

do  the  four  contain  ?  ^  Ans.  188A.  2R.  20P. 

20.  A  jeweller  has  25  gold  rings  weighing  S^-dwt.  each;  how 
many  oz.  do  they  all  weigh  ?  Ans.  4|oz. 

21.  Bought  22gal.  of  molasses  at  75ct.,  2471b.  of  sugar  at 

16|ct.,  1751b.  of  rice  at  6}ct.,  and  57-^-lb.  of  coffee  at 
18  ^ct.;  what  was  my  bill  ? 

22.  Find  the  value  of  25000-4-125  +  1475^25.      Yal.  259. 

23.  What  is  the  quotient  of   three  fourths  by   seventeen 
thirty-thirds?  Ans.  l|-t. 

24.  What  is  the  value  of  17.375-^2.5-9.63-^3.3  ? 

25.  In  392981b.  of  iiour,  how  many  bbl.?         Ans.  200-^bbl. 

26.  A  father  divided  778A.  3R.  2 IP.   equally  among  his 

seven  children  ;  what  was  each  child's  share  ? 

Ans.  IIIA.  IR.  3P. 

27.  What  is  the  cost  of  273bu.  of  wheat  at  $1.66f  per  bn.? 

28.  What  is  the  cost  of  29rra.  of  paper  at  $2.75  per  rm.? 

Ans.  $79.75. 

29.  What  part  of  lOda.  is  7hr.  15min.?  Ans.  ■.%%. 

157 


§209  PROMISCUOUS   PROBLEM ij. 


30.  What  part  of  5gal.  Iqt.  is  3qt.  Sgi.l 

31.  Bought  I  of  17T.  3qr.  of  iron  at  5ct.  per  lb.;  what  did 
I  pay  ?  Ans.  $1363. 

32.  In  31b.  Avoirdupois,  how  many  oz.  Troy  ? 

Ans.  43.75oz. 

33.  What  part  of  lib.  Avoir,  is  lib.  Troy? 

34.  How  many  cubic  inches  in  40qt.  Wine  Measure  ? 

Ans.  2310cu.in. 

35.  What  cost  3doz.  Arithmetics  at  $2.75  per  doz.jl7  slates 

at  I4ct.,  5  gross  of  steel  pens  at  93^}ct.  per  gross,  and 
300  slate  pencils  at  31|ct.  per  hundred  ? 

36.  Bought  156bbl.  of  flour  for  $936,  and  sold  the  same  at 
^8.45  per  bbl.;  what  did  I  gain  ? 

37.  In  Ibu.,  how  many  qt.  Wine  Measure  ? 

Ans.  37.236qt. 

38.  How  much  heavier  is  a  pound  of  feathers  than  a  pound 
of  gold  ?  Ans.  1240gr. 

39.  Virginia  contains  6 I352sq.mi.;  North  Carolina,  55500; 
South  Carolina,  28000;  Georgia,  58000;  Florida, 
59268;  Alabama, 50722  ;  Mississippi,47151;  Louisiana, 
41346;  Texas,  325520;  Arkansas,  52198;  Missouri, 
65037;  Tennessee,  44000;  and  Kentucky,  37680. 
What  is  the  area  of  the  Confederate  States  of  America  ? 

40.  In  lib.  Troy,  how  many  oz.  Avoir.? 

Ans.  13.0281 -foz. 

41.  What  will  33J-yd.  of  cloth  cost  at  $4.75  per  yd.? 

Ans.  $159,125. 

42.  What  will  17bbl.  of  flour  cost  at  $7,875  per  bbl.? 

43.  What  will  60bu.  of  wheat  cost  at  $1,125  per  bu.? 

44.  A  man  borrowed  $189.75,  and  paid  at  one  time  $37,375, 

at  an  other  $23,625,  and  at  an  other  $19.4375  ;   how 
much  does  he  still  owe  ?  Ans.  $109.3125. 

158 


PROMISCUOUS  PROBLEMS.  §209 


45.  A  lady  bought  a  silk  dress  for  $21,875,  a  lace  mantle 
for  $15.50,  a  pair  of  cloth  gaiters  for  $3.25,  and  a 
bonnet  for  $9,625  ;  what  did  they  all  cost  ? 

4Q.  If  25yd.  of  cloth  cost  $85.50,  what  does  1yd.  cost  ? 

*  Ans.  $3.42. 

47.  What  will  3651b.  of  flour  cost  at  $4  per  hundred  ? 

Ans.  $14.60. 

48.  How  many  working  days  are  there  in  a  common  year  ? 

49.  If  a  man  receives  $2000  a  year, how  much  is  that  a  day? 

Ans.  $5,479. 

50.  Bought  5bu.  at  $1.37-^-  per  bu.,  and  sold  them  at  5ct.  per 

qt.  Wine  Measure ;  how  much  did  I  gain  ? 

Ans.  $2,434  +  . 

51.  How  many  steps  of  28in.  each,  does  a  soldier  take  in 

marching  5  miles? 

o2.  How  many  bottles  containing  Iqt.  igi.  each,  can  be  filled 

from  a  hogshead  of  French  Brandy  ? 

Ans.  237.295  bottles. 

53.  If  a  familj^  use  15bb].  of  flour  in  a  year,  how  much  is 

that  a  day  ?  Ans.  8.051b. 

54.  If  a  man  travel  29mi.  Tfur.  I5rd.  per  day,  how  far  will 
he  travel  in  5wk.  if  he  rest  on  Sunday  ? 

55.  A  lady  went  shopping  with  £>^,  and  spent  -}  of   14s.; 

how  much  had  she  left  1  Ans.  6|s. 

50.  How  many  days  are  there  from  Jan.  17  to  April  6  ? 

Ans.  79da. 

57.  Sold  one  load  of  hay  weighing  I.IT.,  an  other  weighing 

1|^T.,  andathird  weighing  17.3c wt.;  what  did  the  three 
loads  weigh  ? 

58.  Bought  }  of  ^  of  an  acre  in  one  lot,  49P.  in  an  other, 

and  -}  of  lOR.  in  an  other ;  what  did  the  three  lots  cost 
at  $69.6875  per  A.?  Ans.  $91,029  + . 

159 


^209  I'RO.MlbCUOUS   PEOBLEMli. 

5*>.   ^V iiat  wiil  ^o.ailb.  of  corn  cost  at  40ct.  per  bu.ir 

Ads.  $18.21. 
GO.  What  will  10001b.  of  wheat  cost  at  $1.37  J-  per  bu.? 
<il.   If  Ibu.  of  wheat  will  make  451b.  of  flour,  how  many> 

bl>l.  will  15001b.  make  ?  Ans.  5.74bbl. 

C>'2.  How  many  secouds  were  there  in  the  winter  of  1850—60? 

Ans.  7S62400sec. 
G3.  How  many  minutes  were  there  in  the  summer  of  1860  ? 
G4.  How  many  acres  of  land  at  $1  per  sq.yd.  can  be  bought 

for  S15000  ?  Ans.  3.099A. 

()5.  What  will  200mi.  of  Tclegrapii  wire  cost  at  lOct.  per 

yd.?  An.s.  $35200. 

66.  How  many  pounds  of  flour  in  75bbl.? 

67.  What  is  the  diiference  in  height  between  a  man  5ft. 
llin.  high,  and  a  horse  16  hands  high  ?  Ans.  7in. 

Os.  Bought  101b.  of  rhubarb  at  $6.50  per  lb.  Avoir.,  and 
sold  it  at  50ct.  pnr  oz.  Troy  ;  what  did  I  gain  ? 

Acs.  87.916. 

69.  What  cost  2127ft.  of  lumber  at  $  .8375  per  hundred  ? 

70.  What  cost  37560  bricks  at  1^7.75  per  thousand  ? 

Ans.  $291.09. 

71.  What  cotit  17  firkins  of  butter  at  18^ct.  per  lb.? 

Ans.  $178.50. 

72.  What  cost  5.5  stone  of  potatoes  at  1.5ct.  per  lb.? 

73.  What  decimal  fraction  is  equal  to  02^-^129  ? 

"  Ann.  .484496  +  . 

74.  A  man  dying  left  $27000  to  be  divided  so  that  his  widow 
should  have  one  third  of  it,  each  one  of  4  sons  one 
seventh  of  the  remainder,  and  each  one  of  5  daughters 
one  fifth  of  what  was  left ;  what  was  each  daughter's 
share  ?  Ans.  $1542.857. 

160 


PROMISCUOUS   PROBLEMS.  §209 

75.  A  druggist  having  bought  60gal.  of  oil  for  $97.50,  lost 

6.25gal.  by  leakage,  and  sold  the  remainder  at  $2,125 
per  gal.:  what  did  he  gain? 

76.  A  merchant  bought  two  bales  of  domestic,  containing 
each  20  bolts  of  38yd.  at  13.25ct.  per  yd.:  what  did  he 
pay  ?  Ans.  $174.90. 

77.  Bought  SOObbl.  flour  at  $6.75  per  bbl.;  sold  one  third 

of  it  at  $7,375  per  bbl.,  one  half  of  the  remainder  at 
$7.9375  per  bbl.,  and  the  rest  at  $8.50  per  bbl.  :  how 
much  did  I  gain  ?  Ans.  $356.25. 

78.  What  will  2250bu.  corn  cost  at  |-  of  a  dollar  per  bu.? 

79.  A  gentleman's  house  cost  him  four  times  as  much  as  his 
furniture,  and  both  together  cost  $4435  ;  what  did  his 
furniture  cost  ?  Ans.  $887. 

80.  A  grocer  had  7cwt.  3qr.  of  sugar,  and  sold  at  different 
times  3icwt.,  3iqr.,  and  1271b.:  how  many  lb.  has  he 
remaining  ?  Ans.  23l^lb. 

81.  A  planter  sold  15  bales  of  cotton  averaging  445.51b.  at 
9ct.  per  lb.,  and  with  the  proceeds  bought  land  at 
$21.25  per  A.:  how  much  did  he  buy  ? 

82.  If  4f yd.  of  cloth  cost  $12|,  what  will  1yd.  cost  ? 

83.  What  cost  29A.  IR.  18P.  of  land  at  $45,625  per  A.? 

Ans.  $1339.664 -f. 

84.  What  cost  9T.  IGcwt.  151b.  of  iron  at  $37.75  per  T.? 

85.  A  man  left  -}  of  his  estate  to  his  wife,  -}  of  the  remain- 
der to  his  son,  and  the  remaining  $2500  to  his  daugh- 
ter :  what  was  his  estate  ?  Ans.  $7500. 

86.  If  16^  days'  work  cost  $19.75,  what  will  3:^  days'  work 

cost'/  Ans.  $3.89  +  . 

S7.  What  must  I  pay  for  6:2lb.  of  butter  at  35ct.  per  lb.,  , 

12vdoz.  eggs  at  15ct.  per  doz.,  10  chickens  at  18fct. 

apiece,  and  30  cucumbers  at  lOct.  per  doz.? 
K  IGl 


§209  pROMiscuoiJS  problems'. 


SS.  If  Ibbl.  of  tar  cost  $3,875,  what  will  l7bM.  cost  ? 

Ans.  $65,875. 

89.  What  cost  2471b.  dried  blackberries,  at  15ct.  per  lb.? 

Ans.  $37.05.- 

90.  What  will  1001b.  of  coffee  cost  at  61b.  to  the  dollar? 

91.  How  many  dollars  will  pay  for  15  pieces  of  French  cal- 

ico^ each  containing  27yd.,  at  1.2fr.  per  yd.? 

Ans.  $90,396. 

'92.  "Howm&ny  dollars  will  pay  for  75  gross  of   G-illott's 

pens  at ^So  6d.  per  gross?  Ans.  $63,526, 

93.  What  will  45bu.  corn  cost  at  5}^  dimes  per  bu.? 

94.  What  will  7271b.  salt  cost  at  $1.25  per  bu.? 

Ans.  $18,175. 

95.  What  will  Sbbl.  flour  cost  at  $.05  per  lb.?    Ans.  $29.40. 

96.  How  many  bu.  of  corn  can  be  hauled  by  a  team  which 
can  haul  just  60bu.  of  wheat  t 

.97.  John's  height  is  3  cubits  and  a  span  ;  his  pony  is  14 
hands  high  ;  what  is  the  difference  of  their  heights  ? 

Ans.  7in. 

98.  How  many  ft.  of  water  is  drawn  by  a  vessel  which  can 
not  sail  in  less  than  3  fathoms  2  feet  ?  Ans.  20ft. 

99.  What  wiiri280cu.ft.  of  wood  cost  at  $1.75  per  cord? 

100.  What  should  be  paid  for  570bu.  of  corn,  at  $2.50  per 
bbl.?  Ans.  $285. 

101.  A  merchant  bought  21  pieces  of  cloth,  each  containing 
41  yards,  for  which  he  paid  $1260  ;  he  sold  the  cloth 
at  $1.75  per  yd.:  did  he  gain  or  lose  by  the  bargain  ? 

Ans.  He  gained  $246.75. 

102.  A  man  receives  }  of  his  income,  and  finds  it  equal  to 
$8724.16  :  how  much  is  his  whole  income  ? 

103.  If  322  books  cost  $371.91,  what  will  248  books  cost  at 
the  same  rate  1  Ans.  $286.44 

162 


KATIO.  §215 


EATIO. 

§  210.  The  ratio  of  one  number  to  an  other  of  the  same 
denomination  is  the  quotient  of  the  second  divided  hy  the 
first.  Thus,  the  ratio  of  3  to  12  is  4 ;  the  ratio  of  5ft.  to 
15ft.  is  3  ;  the  ratio  of  $17  to  $8  is  -V- 

§  211.  Since  the  two  numbers  compared  are  necessarily 
of  the  same  denomination,  every  ratio  is  an  abstract  num- 
ber. (§  185.) 

§  212.  Of  two  numbers  compared,  the  first  is  called  the 
antecedent,  the  second  is  called  the  consequent,  and  both  to- 
fcether  are  called  the  terms  of  the  ratio.  Thus,  in  the  first 
ratio  above,  3  is  the  antecedent,  12  is  the  consequent,  and  3 
and  12  are  the  terms. 

§213.  A  ratio  is  usually  denoted  by  a  colon  placed  be- 
tween the  two  terms.  Thus,  3  :  12  is  the  ratio  of  3  to  12  ; 
so  also,  5ft.  :  15ft.=:3  ;  $17  :  $8=-jV- 

§  214,  The  ratio  of  two  numbers  ef  the  same  nature  but 
of  different  denominations  may  be  found  by  first  reducing 
them  to  the  same  denomination.  Thus,  3ft.  :  5yd.  =  5; 
Oct.  :  $1  =  20. 

§  215.  The  ratio  of  two  numbers  of  different  natures  can 
not  be  found.     Thus,  3ft.  has  no  ratio  to  Sgal. 

Ex.  1.  What  is  the  ratio  of  3  to  6  ?  Ans.  2. 

2.  What  is  the  ratio  of  10  to  75  ?  Ans.  7.5. 

3.  What  is  the  ratio  of  27  to  9  ? 

4.  What  is  the  ratio  of  446  to  1338  ?  Ans.  3. 

5.  What  is  the  ratio  of  $97  to  $485  ?  Ans.  5. 

6.  What  is  the  ratio  of  27qt.  to  9qt.? 

7.  What  is  the  ratio  of  3qt.  to  5gal.?  Ans.  6.6. 

163 


§216  PROPORTION. 


8.  What  is  the  ratio  of  7fur.  to  llmi.?         Ans.  12.57142. 

9.  What  is  the  ratio  of  ^£.5  to  15s.? 

10.  What  is  the  ratio  of  -}  to  {-1  Ans.  1.75. 

11.  What  is  the  ratio  of  3.75  to  11.25  ?  Ans.  3. 

12.  What  is  the  ratio  of  5:^  to  17^  ? 

13.  What  is  the  ratio  of  3ioz.  to  1-Vlb.  Aroir.?       Ans.  6-^. 

14.  What  is  the  ratio  of  45min.  to  ihr.?  Ans.  -i^. 

15.  What  is  the  ratio  pi  1.25cu.ft.  to  2.5cu.in.? 

16.  What  is  the  ratio  of  f  A.  to  15P.?  Ans.  .125. 

17.  What  is  the  ratio  of  Ihhd.  to  25gal.?        Ans.  .3668  +  . 

18.  What  is  the  ratio  of  1.5  cubits  to  65  inches  ? 


SIMPLE  PEOPORTION. 


§216.  A  2'>t'oportion  is  an  equality  of  two  ratios.  Thus, 
the  two  ratios,  5  :  10,  and  Sin.  :  6in.,  form  a  proportion. 

§  217.  A  proportion  is  denoted  by  a  double  colon  between 
the  two  ratios,  or  by  a  sign  of  equality  between  them. — 
Thus,  5  :  10  ::  3in.  :  6in.,  read,  5  is  to  10  as  Sin.  is  to  6in. 
Or,  5  :  10z=:3in.  :  6in.,  read,  the  ratio  of  5  to  10  is  equal 
to  the  ratio  of  Sin.  to  6in. 

§  218.  The  first  two  terms  of  a  proportion  are  called  the 
first  couplet^  the  last  two  are  called  the  second  couplet: — 
the  first  and  third  terms  are  called  the  antecedents;  the 
second  and  fourth  are  called  the  consequents  : — the  first  and 
fourth  terms  are  called  the  extremes ;  the  second  and  third 
are  called  the  means. 

Also,  the  fourth  term  is  called  a  fourth  proportional  to 

the  other  three  :  and,  when  the  means  are  equal  to  each 

other,  either  mean  is  called  a  mean  proportional  between 

the  two  extremes. 

164 


SIMPLE   PROPORTION.  §220 


§  219.  In  every  proportion,  the  product  of  the  extremes 
is  equal  to  the  product  of  the  means. 

For,  if  3  :  6  ::  7  :  14,  6-3  =  14-7,  or  2=  V,  multiply- 
ing both  terms  of  the  first  fraction  by  7,  and  both  terms  of 

the  second  by  3,  we  have  ~~^  ^^,  or  6  x  7=3  x  14. 

§  220.  This  property  enables  us  to  find  any  term  of  a 
proportion  when  the  other  terms  are  given. 

If  the  two  means  and  either  extreme  are  given,  to  find 
the  other  extreme,  we  divide  the  product  of  the  means  by 
the  given  extreme. 

If  the  two  extremes  and  either  mean  are  given,  to  find 
the  other  mean,  we  divide  the  product  of  the  extremes  by 
the  given  mean. 

Ex.  1.  1st  Term  :  6  ::  5  :  15,  what  is  the  first  term  '/ 

Ans.  2. 
2.  7  :  2nd  Term  ::  14  :  70,  what  is  the  second  term  ? 

Ans.  35. 

o.  5-^-  :  22  ::  3rd  Term  :  40,  what  is  the  third  term  ?       ■ 

4.  8  :  1.6  ::  50  :  4th  Term,  what  is  the  fourth  term? 

Ans.  10. 

5.  15  :  1.875  ::  3rd  Term  :  5,  what  is  the  third  term  ? 

Ans.  40. 

6.  2i  :  2nd  Term  ::  7-}  :  12 1-,  what  is  the  second  term  ? 

7.  1st  Term  :  1a  ::  -|  of  a  :  17,  what  is  the  first  term  ? 

Ans.  ^. 
b.  29  :  2nd  Term  ::  17  :  49,  what  is  the  second  term? 

Ans.  83-}-^. 
^.  lo  :  18  ::   3rd  Term  :  24,  what  is  the  third  term? 
10.  2i  :  31  ::  4^  :  4th  Term,  what  is  the  fourth  term  ? 

Ans,  6.3. 
165 


§221  PROPORTION. 


§  221.  Whichever  term  is  required,  however,  the  given 
terms  may  always  be  arranged  in  the  first,  second,  and  third 
places,  so  that  the  work  shall  consist  in  finding  a  fourth 
proportional  to  the  three  given  terms.     Thus, 

The  first  question  would  become,  15  :  5  ::  G  :  4th Term; 

The  second,  14  :  70  ::  7  :  4th  Term  ; 

The  third,  12  :  5^  ::  40  :  4th  Term. 

In  finding  a  fourth  proportional  to  three  concrete  num- 
bers, the  first  two  terms  must  be  of  the  same  denomination, 
and  this  common  denomination  must  be  canceled  before  the 
operation  is  performed!  In  speaking,  hereafter,  of  the  first 
qr  the  second  term,  we  will  always  mean  the  number  of  ab- 
stract units  in  such  term. 

Ex.  11.  Find  a  fourth  proportional  to  3in.,  7in.,  and  $12. 
3  :  7  ::  $12  : 


p^2.  Model.— Multiply  the 

'-  third  term  by  the  second.  (§183). 

3)$84  Divide  the  product  by  the  first 

$28  term.     (^185).     The  fourth  term 

is  $28. 

Explanation. — The  necessity  of  considering  the  first 
two  terms  abstract,  is  evident  from  the  fact  that  $12  can 
not  be  multiplied  by  Tin.,  neither  can  $84  be  divided  by  Bin. 

liULE. — MuIfqyJj/  the  third  term  hy  the  second,  and  divide 
the  2^'^oduct  hy  the  first. 

Or,  Multiply  the  third  term  hy  the  ratio  of  the  first  to  the 
second. 

Ex.  12.  Find  a  fourth  proportional  to  15yd.,  25yd.,  and 
lOda. 

13.  Find  a  fourth  proportional  to  $5,  $75,  and  $40. 

14.  Find  a  fourth  proportional  to  7da.,  15da.,  and  £2,  63. 
9d.  4th  Term,  £5,  2}d. 

16G 


SIMPLE   PROPORTION,  §22o 


15.  Find  a  fourth  proportional  to  5|-qt.,  14qt.,  and  $17.75. 

16.  Find  a  fourtla  proportional  to  3pk.,  2bu.,  and  $7.50. 

4th  Term,  S20. 

17.  Find  a  fourth  proportional  to  1,1^  43,  and  14s. 

4th  Term,  7s.  9d.  S-^^qr, 

18.  Find  a  fourth  proportional  to  3f,  7f,  and  lOgal.  Sqt, 

19.  Find  a  fourth  proportional  to  lOgal.,  3qt.,  and  5.5da. 

4th  Term,  .4125da. 

20.  Find  a  fourth  proportional  to  3s.,  lOda.,  and  5yd. 

4th  Term,  1.3$yd. 

21.  If  15bu.  of  corn  cost  $10,  what  will  27bu.  cost? 

15  :  27  ::  $10  :  $18 

27  §223.   Model.— 110  is  the 

%^1^  15        third  term.     Since  the  cost  of 
27bu.  is  2;reater  than   that  of 


120 


-l$18  i5bu.^  15  is  the  first  term,  and 
27  the  second.  Multiply  the 
third  term  by  the  second.  (§  183,)  Divide  the  product  by 
the  first  term.  (§  185.)     Hence,  27bu.  cost  $18. 

Explanation. — The  ratio  of  15bu.  to  27bu.  is  evidently 
the  same  as  the  ratio  of  the  value  of  15bu.  to  the  value  of 
27bu.,  that  is,  the  ratio  of  $10  to  the  required  amount. 
Hence,  the  propriety  of  the  proportion.  The  first  and  sec- 
ond terms  are  considered  abstract  from  the  necessity  of  the 
case. 

KuLE. —  Take  for  the  third  term  the  given  number  ivhich 
is  of  the  same  nature  with  the  required  term :  and,  if  the  re- 
quired term  is  evidently  greater  than  this  third  term,  take  the 
greater  of  the  remaining  terms  for  the  second  and  the  less  for 
the  first  ;  hut,  if  the  required  term'  is  less  than  the  third  term, 
take  the  less  of  the  remaining  terms  for  the  second  and  the 
grtater  for  the  first. 

Find  a  fourth  proportional  to  the  three  terms  thus  arranged, 

167 


$223  PROPORTION. 


Ex.  22.  If  101b.  of  sugar  cost  $1.25,  what  will  17.51b. 
cost?  Ans.  $2.1875. 

23.  If  27bbl.  of  flour  cost  $150.75,  what  will  94.5bbl.  cost  'i 

Ans.  $527,625. 

24.  What  cost  llgal.  of  molasses,  if  49gal.  cost  $34.47? 

25.  What  cost  3.75A.  of  land,  if  16.375A.  cost  $400  ? 

Ans.  $91.60  +  . 

26.  If  12  horses  eat  a  load  of  hay  in  10  days,  how  long 
would  a  load  last  5  horses  ?  Ans.  24da. 

27.  If  12  horses  eat  a  load  in  10  days,  how  many  horses 
would  eat  it  in  5  days  ? 

28.  If  91b.  of  tea  cost  $10,  what  will  111b.  cost  ? 

29.  If  a  6  penny  loaf  of  bread  weigh  5.5oz.  when  flour  is 

$4.50  per  bbl.,  what  should  it  weigh  when  flour  is  $7 
per  bbl.?  Ans.  3.535oz. 

30.  If  -|  of  a  yard  of  cloth  cost  $2.75,  what  will  -^-^  of  a 
yard  cost? 

31.  What  cost  1751b.  of  cofi'ee,  at  5-i  lb.  for  a  dollar  ? 

Ans.  $31.81  +  . 

32.  What  cost  45  pr.  of  shoes,  if  14  pr.  cost  $35.50  ? 

Ans.  $114,107  +  . 

33.  If  $100  gain  $6  in  lyr.,  in  how  many  years  will  it  gain 
$100? 

34.  If  12  men  build  20rd.  of  masonry  in  a  week,  how  many 
men  could  build  75rd.  in  the  same  time  ?  Ans.  45  men. 

35.  If  lObbl.  of  flour  will  last  a  company  of  soldiers  ISda., 
how  long  will  lOOOIb.  last  them  ?  Ans.  7.653da. 

36.  If  I4lb.  of  butter  cost  $4.25,  what  will  1.75  firkins  cost  ? 

37.  If  a  boat  travels  75mi.  in  6hr.,  how  far  does  it  go  in 
25min.?  Ans.  5mi.  Ifur.  26.6rd. 

38.  If  87.51b.  of  coflfee  cost  M,  12g.  6d.,  what  will  7.51b. 
cost?  Ans.  7s.  lid. 

168 


COMPOUND   PROPORTION.  §224 

39.  If  a  man  can  walk  lOmi.  in  3hr.,  how  far  can  he  walk 

in  5da.  of  8hr.  each  ? 

40.  If  5.51b.  of  sugar  cost  $1.00,  what  will  lib.  cost? 

Ans.  $  .18. 

41.  If  5ibu.  of  wheat  make  Ibbl.  of  flour,  how  much  flour 
will  25001b.  of  wheat  make  ?  Ans.  7.5'7bbl. 

42.  If  l^gal.  of  molasses  cost  $1.29,  what  will  l-^hhd.  cost? 

43.  If  lOijd.  of  cloth  cost  $11,625,  what  will  16|-yd.  cost? 

Ans.  $18,452  +  . 

44.  Iff  of  a  ship  cost  ^500,  7s.  3d.,  what  will  -j^^-  of  her 

co«t?  Ans.  £260,  12s.  1.3125d. 

45.  If  3  reams  of  paper  cost  $7.75,  how  many  reams  can  be 

bought  for  $17.75  ? 

46.  How  many  lb.  Avoir,  are  equal  to  500  lb.  Troy? 

47.  A  grocer  was  detected  in  using  as  a  gallon  measure  a 

vessel  containing  3qt.  Ipt.  2^gi. :  how  many  true  gal- 
lons were  in  47.5  of  his  false  gallons?    A.  45.273  + gal. 

48.  The  same  man  used  for  his  purchases  a  vessel  contain- 
ing 4qt.  2gi. :  how  many  true  gallons  were  in  47.5  of 
these  false  gallons  ? 

49.  How  many  of  his  selling  gallons  were  in  47.5  of  his 
buying  gallons?  Ans.  52.95  +  . 

50.  If  1001b.  of  gunpowder  require  751b.  of  saltpetre,  how 

much  saltpetre  will  22.251b.  of  gunpowder  require  ? 


COMPOUND  PROPOKTION. 


§  224.  A  compound  ratio  is  the  product  of  two  or  more 

simple  ratios. 

3  '  5  ) 
Thus,    .  '.  ^,  >  ::  12  :  35,  is  a  compound  ratio. 

169 


§225  PROPORTION. 


§  225.  A  compound  proportion  is  an  expression  of  equal- 
ity between  a  compound  and  a  simple  ratio. 

§  226.  A  compound  ratio  is  reduced  to  a  simple  one,  by 
multiplying  together  its  corresponding  terms.  Thus,  in  the 
above  instance,  the  antecedent  12  is  the  product  of  the  an- 
tecedents 3  and  4,  and  the  consequent  35  is  the  product  of 
the  consequents  5  and  7. 

§  227.  If  the  first  three  terms  of  a  compound  proportion 
are  given,  the  fourth  term  may  be  found  by  multiplying  the 
third  term  by  the  product  of  the  second  terms,  and  dividing 
this  product  by  the  product  of  the  first  terms  :  the  first  and 
second  terms  being  reduced  to  the  same  denomination  in 
each  simple  ratio,  and  then  considered  abstract. 

Ex.  1.  If  5  hands  can  hoe  24A.  of  cotton  in  4da.,  how 
many  Acres  can  17  hands  hoe  in  llda.? 

^  •   ^^^  ••   24A    •  224^  \         ^^^^-  Model.— 24A. 
4  :   11  ^  "  *  *     "*  5-  •     is  the  third  term.    Since 

20     137  17  hands  can  hoc  more 

2^^^  than  5  hands,  5  is  a  first 


748 
374 


term,  and-  17  a  second. 
Since  more  land  can  be 
hoed  in  llda.  than  in  4 
2,-0)448,8A.  (la^^  4  jg  ^  first  term  and 

224|-A.  11  a  second.     4  times  5 

are  20.  11  times  17  are 
187.  Multiply  the  third  term  by  187.  (§  183.)  Divide  the 
product  by  20.  (§185.)  Hence  the  required  number  is 
224|A. 

Explanation.— As  in  simple  proportion,  we  take  for  the 
third  term  that  which  is  of  the  same  nature  with  the  re- 
quired term.  The  remaining  numbers  are  in  pairs  of  sim- 
ilar terms,  and  each  pair  is  arranged  ^s  if  the  question  de- 
pended upon  it  alone. 

170 


COMPOUND   PROPORTION.  §^8 


Rule. —  Take  for  the  third  term  the  given  number  which 
is  of  the  same  natvre  with  the  required  term.  Take  the  re- 
maining numbers  in  pairs  of  the  same  nature,  and  arrange 
each  pair  as  in  simple  proportion. 

Multiply  the  third  term  by  the  prodiict  of  the  second  termSy 
and  divide  this  pj'oduct  by  the  product  of  the  first  terms. 

Ex.  2.  If  60  men  can  do  a  piece  of  work  in  40  days  of  8 
hours  each,  how  many  men  can  do  three  times  the  work  in 
90  days  of  10  hours  each  %  Ans.  04  men. 

3.  If  10  horses  eat  88bu.  of  corn  in  45  days,  how  many 
horses  will  eat  120bu.  in  50  days? 

4.  If  I  can  travel  75  miles  in  2ida.  of  7-J-hr.  each,  how 

many  da.  of  10  lir.  each  would  it  take  me  to  travel 
225mi.?  Ans.  5|da. 

5.  If  5  white  men  can  do  as  much  work  as  7  negroes,  how 
many  days  of  lOhr.  each  will  be  required  for  25  negroes 
to  do  a  piece  of  work  which  30  white  men  can  do  in  10 
days  of  9hr.  each  r*  Ans.  15,12da: 

6.  If  by  traveling  7hr.  per  da.  at  4nii.  per  hr.  I  go  280mi. 

in  lOda.,  how  far  will  I  go  in  12da.  by  traveling  8hr. 
per  da.  afc  4^mi.  per  hr.? 

7.  If  12  men  build  9  rods  of  wall  in  10  days,  how  many 

men  can  build  27  rods  in  5  days  ?  Ans.  72  men. 

8.  If  $100  gain  SfjO  in  12mo.,  what  will  ^375  gain  in  9mo.? 

Alls.  $16,875. 

9.  If  1001b.  be  carried  lOOini.  for  35ct.,  what  will  be  the 
freight  on  100001b.  carried  75mi.? 

10.  If  9  men,  in  10  days,  can  build  a  v.all  25rd.  long,  3yd. 

higl),  and  5ft.  thick,  in  what  time  can  20  men  build  a 
wall  30rd.  long,  4yd.  high,  and  4ft.  thick  ?    A.  5.76da. 

11.  If  100  men,  in  5da  of  lOlir.  each,  can  dig  a  ditch  150 
yd.  long,  .3yd.  wide,  and  5ft,  dec;',  how  many  men,  in 

171 


,^2^9  PROPORTION. 


8da.  of  9hr,  each,  can  dig  a  ditch  200yd.  long,  S^-yd. 
wide,  and  2yd.  deep?     A.  omitting  fraction,  123  men. 

12.  If  $100  gain  $7  in  12ino.,  in  how  many  months  will 
•   $700  gain  $100? 

13.  If  $100  gain  $8  in  12mo.,  what  sum  of  money  will  gain 
$160  in  ISmo.  ?  Ans.  1333.33i 

14.  If  $900  gain  $135  in  18mo.,  what  will  $100  gain  in 

12mo.  ?  Ans.  $10. 

15.  If  6  school-girls  spend  $72  pocket-money  in  4wk.,  what 

will  10  girls  spend  in  6wk.  ? 

16.  If  900  soldiers  eat  70bbl.  of  flour  in  20da.,  how  many 
days  will  200bbl.  last  3000  soldiers  at  half  rations  ? 

Ans.  34-2-da. 


PARTITIVE  PROPORTION ;   OR,  FELLOWSHIP. 


§  229.  Partitive  Proportion  is  the  division  of  a  num- 
ber into  two  or  more  parts  which  shall  have  to  each  other  a 
given  ratio. 

The  terms  of  the  ratio  are  called  the  proportional  terms, 
and  in  the  operation  they  must  be  regarded  abstract. 

Ex.  1.  Divide  450  into  three  parts,  which  shall  be  to 
each  other  as  2,  3,  and  4. 

9  :  2  ::  450  :  100  §  230.  Model.--2  and  3  are  5, 
9:3::  450  :  150  and  4  are  9.  9  is  to  2  as  450  is 
9  :  4  ::  450  :  200     to  the  first  part.     Multiply  450 

by  2.  Divide  the  product  by  9. 
The  first  part  is  100.  9  is  to  3  as  450  is  to  the  second  part. 
Multiply  450  by  3.  Divide  the  product  by  9.  The  second 
part  is  150.  9  is  to  4  as  450  is  to  the  third  part.  Multiply 
450  by  4.  Divide  the  product  by  9.  The  third  part  is  200. 
Hence,  100,  150,  and  200,  are  the  three  parts  required. 

172 


TARTITIYE   rROPORTION.  §230 


Explanation. — One  half  of  the  first  part  is  evidently 
equal  to  one  third  of  the  second  or  one  fourth  of  the  third  : 
so  that  if  the  whole  number  be  divided  into  9  equal  parts, 
the  first  will  contain  2,  the  second  3,  and  the  third  4,  of 
those  parts.     Hence  the  truth  of  the  proportions. 

Rule. — As  the  sum  of  the  proportional  terms  is  to  cither 
term,  so  is  the  whole  number  to  be  divided  to  the  part  corre- 
sponding to  that  term. 

Proof. — Add  the  several  parts  together  :  their  sum  is 
equal  to  the  whole  number  divided. 

Ex.  2.  Divide  $1000  into  three  parts  which  shall  be  to 
each  other  as,  6,  1,  and  3.  $600,  $100,  and  $300- 

3.  Divide  141A.  into  three  parts  in  the  proportion  of  1^, 

3i,  and  4}. 

4.  Divide  226gal.  into  four  parts  in  the  proportion  of  3-^., 

2-|-,  2i,and  2-^-.  42gal.,  56gal.,  GOgal.,  and  68gal. 

5.  Divide  992.95  into  four  parts  in  the  proportion  of  1.25, 

3.2,  4.73,  and  5.005.  87.5,  224,  331.1,  350.35. 

6.  Divide  $43.20  into  four  parts  in  the  proportion  of  1,  3, 

5,  and  7. 

7.  Two  men,  A  and  B,  engage  in  business  with  a  joint 
capital  of  $6000,  of  which  A  furnishes  $2500  and  B 
the  remainder.  What  is  each  one's  share  of  a  gain  of 
$1200  ?  Ans.  $500,  $700. 

8.  Two  men,  C  and  I),  gain  $1275  on  a  capital  of  which 

C's  share  is  double  D's ;  what  is  each  one's  share  of 
this  gain?  Ans.  $850,  $425. 

9.  If  E  invests  $2375,  and  F  $3225,  and  the  firm  loses 
$700,  how  much  must  each  partner  lose  ? 

10.  A  invests  $3000,  and  B  83500,  in  a  certain  business, 
in  which  the  first  year  they  lose  $325.     After  paying 

178 


^230  PROPORTION. 


this  loss  from  the  funds  of  the  firm,  they  take  in  C  us 
a  partner  with  a  capital  of  $4000,  and  the  second  year 
the  new  firm  gains  ^2035.  Hov/  much  of  this  gain  is 
due  to  each  partner  ?    Ans.  A,  $570;  B,  $665;  C,  1800. 

11.  D  and  E  form  a  partnership  for  two  years,  I)  contribut- 

ing $5000,  and  E  $1750.  The  first  year  they  gain 
$1350.  D  spends  his  share  of  the  gain,  but  E  leaves 
his  share  among  the  funds  of  the  firm.  The  second 
year  they  gain  $2130.  What  is  each  partner's  share 
of  this  last  gain  '/  Ans.  D,  $1500  ;  E,  $630. 

12.  Messrs.  Jones,  Smith,  and  Brown  gained  $5000;  what 
is  each  man's  share  of  this  gain,  if  Smith  owns  twice 
as  much  of  the  capital  as  Brown,  and  Jones  as  much 
as  Smitli*  and  Brown  together  ? 

13.  In  a  certain  firm,  A  owns  1^^,  times  as  much  stock  as  B, 
C  owns  1^-  times  as  much  as  A,  and  D  owns  1-^  tii^ea 
as  much  as  C:  A's  gain  on  a  year's  transactions  is 
8500  ;  what  is  the  gain  of  each  of  the  other  partners  ? 

Ans.  B,  $400  ;  C,  $600;  D,  $700. 

14.  A  merchant  owes  one  creditor  $2000,  and  an  other 
$3500  :  having  failed,  he  can  pay  them  both  only 
$4015  :  how  much  should*  each  creditor  receive  1 

Ans.  $2555,  and  $1460. 

15.  A  man  dying  wills  to  one  son  $2000,  to  an   other  son 
'     $1500,  and  to  his  daughter  $1250  ;   but  after  paying 

his  debts  his  executor  has  in  his  hands   only  $3000. 
How  much  should  he  pay  to  each  legatee  ? 


16.  Three  partners,  A,  B,  and  C,  invest  as  follows : — A  in- 
vests $500  for  2  months ;  B,  $400  for  3  months ;  and  C, 
$300  for   5  months.     They  gain   $740.     What  ought 

each  partner  to  receive  ? 

174 


PARTITIVE   PROPORTION.  §;:;U 


500  X  2  ==  1000        8700  :  1000 
400  X  3  rr:  1200         8700  :  1200 


$740  :  $200 
$740  :  $240 
$740  ;  $300 


.300  X  5^:1500        8700  :  1500 
3700 
§231.  Model.— Twice  500  are  1000.     3  times  400  nrc 
1200.     5  times  300  are  1500.     The  sum   of  tbese  propor- 
tional parts   is  3700.     [Proceed   as  in   §  230.]     Hence  A 
ought  to  receiye  $200,  B,  $240,  and  C,  ^300. 

Explanation. — A's  investment  of  $500  for  2  months  is 
equal  to  an  investment  of  twice  $500,  or  $1000,  for  1  month ; 
so  B's  $400  for  3  months  is  equal  to  3  times  $400,  or  $1200, 
for  1  month  ;  and  C's  ^300  for  5  months  is  equal  to  5  times 
$300,  or  $1500,  for  1  month.  The  several  investments, 
being  thus  referred  to  the  same  unit  of  time,  evidently  fur- 
nish equitjxble  proportional  terms. 

This  is  an  example  of  what  is  called  Compound  Fellow- 

SHIV. 

Ex.  17.  A  firm  of  two  partners  gained  $1750  :  what  was 
each  partner's  share  of  the  gain,  if  A  contributed  $3000  for 
10  months,  and  B  $2500  for  1  year  ? 

Ans.  A's,  $875  ;  B's,  $875. 

18.  A,  B,  C,  and  D,  rented  a  pasture  for  $100.  A  kept  20 
head  of  cattle  in  it  6  months;  B  kept  25  head  5  months; 
C  kept  30  head  5^-  months ;  and  D  kept  50  head  3 
months.    What  part  of  the  rent  ought  each  man  to  pay  ? 

19.  In  a  certain  partnership  A  contributed  $3000  Jan.  1st; 
B  contributed  $2500  Feb.  1st;  and  C  contributed 
$4000  May  1st.  On  the  1st  of  August,  they  lost  by 
fire  $4000.  What  part  of  the  loss  did  each  partner 
sustain  1  Ans.  A,  $1750  ;  B,  $1250  ;  C,  $1000. 

20.  Three  partners  in  trade  gained  $3008  after  15  months' 

business.     A  put  in  $1000  at  first,  and  $2000  3  months 

175 


§232  PROMISCDOUS    PROBLEMS. 


afterwards ;  B  put  in  at  first  $4000,  but  took  out  $2000 
6  months  afterwards ;  and  C  put  in  $2000  at  the  end 
of  5  months,  and  $2000  5  months  afterwards.  What 
was  each  partner's  share  of  the  gain  ? 

Ans.  A's,  $1092  ;  B's,  $1176;  C's,  $740. 


■«»> 


PROMISCUOUS  PPtOBLEMS. 


8bu. 
5 
4 
45 


J.  If  8bu.  of  wheat  are  worth  as  much  as  15bu.  of  corn, 
and  5bu.  of  corn  as  much  as  2cwt.  of  hay,  and  4cwt.  of  hay 
are. worth  $6;  how  many  bu,  of  wheat  can  be  bought  for 

S45  ? 

15       45.4.5.8bu.=7200bu.         ^  232.,  Model.— 
2         6.2.15r=180  Set  8bu.  on  the  left, 

6       7200bu.^l80  =  40bu.    and  15  on  the  right; 

5  on  the  left,  and  2 
on  the  right ;  4  on 
the  left,  and  6  on  the  right,  and  45  on  the  left.  Multiply 
together  the  terms  on  the  left.  Multiply  together  the 
terms  on  the  right.  Divide  7200bu.  by  180.  The  quo- 
tient is  40bu.     Hence  $45  will  pay  for  40bu.  of  wheat, 

Explanation. — This  question,  commonly  referred  to  a 
distinct  head,  called  Conjoined  Proportion,  or  the  Chain 
Rule,  is  merely  a  complicated  case  of  simple  proportion,  as 
will  be  seen  by  stating  it  thus : — 

1.  If  4cwt.  hay  cost  $6,  how  many  cwt.  will  cost  $45  1 

6  :  45  ::  4cwt.  :  30cwt.        Hence,  SOcwt.  hay =$45. 

2.  If  2cwt.  hay=z5bu.  corn,  how  many  bu.  corn  =  80cwt.  hay? 
2  :  30  ::  5bu.  :  75bu.  Hence,  75bu.  corn=$45. 

3.  If  Sbu.  wheat=15bu.  corn,  how  many  bu.  wheat=75bu. 

corn? 
15  :  75  ::  Sbu.  :  40bu.        Hence,  40bu.  wheat=$45. 

176 


CONJOINED  PROPORTION.  §232 


Comparing  this  work  with  the  model,  we  see  that  the 
me<ans  in  the  proportions  are  45,  4,  (30,)  5,  (75,)  and  8 ; 
and  the  extremes,  except  the  last,  are  6,  (30,)  2,  (75,)  and 
15 ;  and  that,  omitting  the  two  terms,  30  and  75,  common 
to  both,  we  have  left  in  the  one  case  the  terms  on  the  left 
in  the  model,  and  in  the  other  the  terms  on  the  right.  And 
since  the  product  of  the  means  ia  equal  to  the  product  of 
the  extremes,  the  product  of  the  terms  on  the  left  divided 
by  the  product  of  the  terms  on  the  right  will  give  the  re- 
«^uired  term. 

The  term  similar  to  the  required  term  is  called  the  odd 
terra,  and  the  one  equivalent  to  the  required  term  is  called 
the  term  of  demand.     Both  of  these  must  be  placed  on  the 
left  of  the  vertical  line,  and  the  other  terms  must  be  arrang- 
ed so  that  equivalents  shall  be  opposite  each  other,  and  no 
two  similar  terms  on  the  same  side.     In  the  operation,  all 
but  the  odd^term  must  be  regarded  abstract. 
t.  If  2bbl.  of  flour  are  worth  as  much  as  26fbu.  of  corn, 
and  3bu,  of  corn  as  much  as  7-^-lb.  of  bacon,  how  many 
lb.  of  bacon  are  equivalent  to  3bbl.  of  flour? 

Ans.  1001b. 
S.  If  £93  are  cqurJ  to  2420fr.,  and  166|fr.   are  equal   to 
$31,  and  87  are  C(]ual  to  4bu.  of  wheat,  how  many  bu. 
of  wheat  are  equal  to  j£15  ? 

4.  If  A  can  do  as  much  work  in  5  days  as  B  can  do  in  6, 
B  can  do  as  much  in  7  days  as  C  can  do  in  8,  and  C 
can  do  as  much  in  9  days  as  D  can  do  in  10 ;  in  how 
many  .days  can  A  do  as  much  as  D  can  do  in  15  ? 

Ans.  9.84da. 

5.  If  10  Ells  Flemish  are  equal  to  6  Ells  English,  and  4 

Ells  English  to  5  yards,  and  12  yards  to  8  Ells  French  ; 

L  177 


§282  PROMISCUOUS  PROBLEMS. 

how  many  Ells  French  are  equal  to  16  Ells  Flemish? 

Ans.  8  E.  Fr. 

6.  If  191b.  of  butter  are  worth  301b.  of  cheese,  and  191b. 
of  cheese  are  worth  $3 ;  how  raacy  lb.  of  butter  are 
worth  S7.50  ? 

7.  If  a  train  of  cars  travel  a  mile  in  2.5min.,  how  long  will 

it  be  in  going  45  miles  ?  Ans.  Ihr.  52.5min. 

$.  If  8  men  can  mow  a  meadow  in  10  days  of  I3hr.  each, 
in  how  many  days  of  llhr.  each  can  12  men  mow  it? 

Ans.  17.'72da. 

9.  A,  B,  and  C  formed  a  partnership  for  two  years  :  th« 
'first  year,  they  lost  $500  ;  the  second  year,  they  gained 
$750 ;  how  much  is  each  partner  entitled  to  at  the 
end  of  the  second  year,  if  A  contributed  $4000,  and  B 
and  C  $3000  each  to  the  funds  of  the  firm  ? 

10.  A  and  B  formed  a  partnership  for  two  years  from  Jan. 

1,  1860.  On  that  day,  A  contributed  $1000,  and  B 
$500  :  July  I,  1860,  A  added  $500  to  his  investment, 
and  Oct.  1,  1860,  B  added  $500  to  his  :  Jan.  1,  1861, 
A  withdrew  $250  from  the  funds  of  the  firm,  and  Mar. 
1,  1861,  B  contributed  $500  more.  They  gain  $1090. 
How  much  of  this  ought  each  partner  to  receive  ? 

Ans.  A,  $600  ;  B,  $490. 

11.  If  5bbl.  of  cider  are  worth  8bu.  of  wheat,  and  11  bu.  of 

wheat  are  worth  2T.  of  coal,  and  3T.  of  coal  are  worth 
501b.  of  tea,  and  41b.  of  tea  are  worth  5oz.  of  quinine, 
and  7oz.  of  quinine  are  worth  $6.50;  how  many  dol- 
lars are  lObbl.  of  cider  worth  ?  Ans.  $56,277. 

12.  If  the  transportation  of  1001b.  lOOmi.  cost  $2.15,  what 

will  it  cost  to  transport  25001b.  25mi.? 

13.  What  is  the  smallest  numbeu.  that  can   be  exactly  di- 

vided by  either  .12,  13,  or  14  I  Ans.  1092. 

178 


PROMISCUOUS  PROBLEMS.  §232 


14.  What  is  the  largest  number  that  will  exactly  divide 
either  240,  720,  or  840 1  Ans.  120. 

1 ').  What  is  the  total  cost  of  4yd.  of  silk  at  $1,875  per  yd., 
Syd.'of  berege  at  ^.625  per  yd.,  3doz.  buttons  at  $.75 
per  doz.,  and  7.5yd.  of  calico  at  $.25  per  yd.? 

16.  What  is  the  total  cost  of  51b.  of  tea  at  5s.  6d.,  7bu.  of 
corn  at  4s.  4d.,  8bu.  of  wheat  at  lis.  9d.,  and  llgal.  of 
molasses  at  7s.  3d.?  Ans.  j^ll,  Us.  7d. 

17.  What  interval  elapsed  between  Dec.  5,  1813,  and  Mar. 
17,  1842  ?  Ans.  28yr.  3mo.  12da. 

18.  What   interval  elapsed  between  Jan.  30,    1833,   and 

Sept.  3,  1862  ? 

19.  John  Jones  was  born  Mar.  9,  1827,  and  was  married 

when  he  was  22yr.  6mo.  lOda,  old  ;  when  was  he  mar- 
ried ?  Ans.  Sept.  28,  1849. 

20.  If  5  men  can  plough  47-^1  acres  in  7|-  days,  in  how 
many  days  can  G  men  plough  31  acres  ?     Ans.  4  days. 

21.  If  o^cwt.  of  hemp  cost  ^27.50,  how  much  hemp  will 
cost  $33.33i  ? 

22.  A  merchant  bought  SOOObu.  of  salt :  after  having  sold 

to  A  lOO.Sbu.,  to  B  477.75bu.,  to  C  329.8375bu.,  and 
to  D  1200. 25bu.,  bow  much  has  he  left  ? 

Ans.  891.6625bu. 

23.  What  is  the  produce  of  15.375A.  of  corn,  at  8bbl.  3bu. 

3pk.  to  the  acre  ?  Ans.  134bbl.  2bu.  2pk.  5qt. 

24.  A  field  of  25  acres  produced  637.5bu.  of  wheat ;   how 

much  was  that  per  acre  ? 

25.  A  sum  of  money  divided  equally  among  17  men  gives 

to  each  ^17.765 ;    if  divided  equally  among  11  men, 
how  much  would  each  get?  Ans.  S27.455. 

26.  If  20  men  in  35da.  earn  $320,  how  many  men  will  earn 
$480  in  70da.?  Ans.  15  men. 

179 


§233  PROPORTION. 


27.  If  18  horses  eat  lObu.  of  oats  in  20(]a.,  how  many  horses 
will  eat  60bu.  in  36cla.? 

28.  What  is  the  greatest  common  measure  of  75,  825,  and 
1575  ?  Ans.  75. 

29.  What  is  the  least  common  multiple  of  46,  230,    and 
115?  Ans.  230. 

30.  What  are  the  different  prime  factors  of  24400  ? 


PERCENTAGE. 


§233.  Percentage  includes  :''•  cases  of  proportion  in 
whichthe  first  term  is  one  liund.   ■■'. 

The  phrase,  »er  centum,  that  is,  ptr  hundred,  is  usually 
written,  and  often  pronounced,  ^er  cent.  Thus,  in  stead  of 
"six  dollars  per  hundred,"  we  usually  say  "6  per  cent." 

Ex.  1.  A  lawyer  collected  $3725  ;  what  is  his  commis- 
sion at  3  per  cent.? 

100  :  3725  ::  $3  :  $112.75  This  proportion  is  evi- 
dently correct :  and  all 
similar  problems  may  be  solved  in  the  same  manner.  But, 
inasmuch  as  the  three  given  terms  have  always  the  same 
unit,  the  same  result  will  be  obtained  by  regarding  the  sec- 
ond term  concrete  and  the  first  and  third  abstract,  by  di- 
viding the  third  by  the  first  and  multiplying  the  second  by 
this  quotient.  This  method,  being  a  little  less  troublesome, 
is  the  one  usually  adopted.  To  explain  it  more  fully,  we 
must  give  the  following  definitions. 

§234.  The  price  or  amount  per  hundred  is  called  the 
rate  per  cent.  Thus,  in  the  above  example,  3  is  the  rate 
per  cent. 

180 


PERCENTAGE.  §238 

§  235.  If  the  rate  per  cent,  be  divided  by  100,  the  quo- 
tient is  called  the  rate  per  unit.  In  all  operations,  this  is 
regarded  as  an  ab.'^tract  number.  Thus,  .03  is  the  rate  per 
unit  in  the  example  above. 

What  is  the  rate  per  unit  for  6  per  cent.?  for  10  per  cent.? 
for  50  per  cent.?  for  75  per  cent.?  for  li  per  cent.]  for  -J- 
per  cent.?  for  100  per  cent.?  for  33^^  per  cent.?  for  \2\  per 
cent.?  for  18 ^^  per  cent.?  for  \  per  cent  ?  for  -^  per  cent.? 

§  236.  The  number  on  which  percentage  is  calculated,  is 
called  the  basis  of  percentage.  Thus,  above,  $3725  is  the 
basis. 

§  237.  The  result  of  the  operation  is  called  the  percent' 
age.     Thus,  above,  $112.75  is  the  percentage. 

Ex.  2.  Find  5  per  cent,  of  $5750. 

$5750         §  238.  Model.— Multiply  the  basis  by  the 

1^     rate  per  unit.  (§  183.)     The  product  is  ,^287.- 

8287.50     50.     Hence  the  percentage  is  ^287.50. 

Explanation. — 5  per  cent,  of  any  number  is  evidently 
5  hundredths  of  that  number,  and  this  is  found  by  multi- 
plying the  number  by  .05.  Observe  that  the  rate  per  unit 
is  simply  one  hundredth  of  the  rate  per  cent.,  and  is  most 
conveniently  expressed  as  a  decimal  fraction. 

KuLE. — Multiply  the  basis  by  the  rate  per  unit.  The 
product  vnll  be  the  percentage, 

Ex.  3.  What  is  1  per  cent,  of  7500  ? 


4.  What 

5.  What 

6.  What 

7.  What 

8.  What 

9.  What 


s  2  per  cent,  of  250  ?  Ans.  5. 

s  3  per  cent,  of  275  ?  Ans.  8.25. 

s  4  per  cent,  of  775  ? 

s  6  per  cent,  of  $325  ?         Ans.  $19.50. 
s  7  per  cent,  of  89250  ?        Ans.  $647.50. 
s  8  per  cent,  of  725  men? 
181 


§239  PROPORTION. 


10.  What  is  9  per  cent,  of  1700  men  ?  Ans.  153  men. 

11.  What  is  -,V  per  cent,  of  $1000  ?  Ans.  61. 

12.  What  is  1  J-  per  cent,  of  8175  ? 

13.  What  is  2^^  per  cent,  of  827.75  ?  Ans.  S.7284375. 

14.  What  is  31  per  cent,  of  8630  ?  Ans.  821. 

15.  What  is  4f  per  cent,  of  795  ? 

16.  What  is  7.}  per  cent,  of  2775.25  ?  Ans.  208.14375. 

17.  What  is  9a  per  cent,  of  473.75  ?  Ans.  46.190625. 

18.  What  is  lO-j-^  per  cent,  of  275  ? 

19.  What  is  16f  per  ce^r.  of  1500  ?  Ans.  250. 

20.  What  is  66f  per  cent,  of  8750  ?  Ans.  $500. 

21.  What  per  cent,  of  690  is  115  ? 

§  239.  Model. — Divide  the  percent- 
age by  the  basis.  (§  52.)    The  quotient 
is  .  16|.   IMultiply  this  quotient  by  100. 
The  product  is   16|.    -Hence,  115   is 
23o'        I-  1C|  per  cent,  of  690. 

l690!3 

ExPLA-NATiON. — Since  the  percentage  is  equal  to  the  basis 
multiplied  by  the  rate  per  unit,  conversely  the  rate  per  unit 
is  equal  to  the  percentage  divided  by  the  basis.  And,  since 
the  rate  per  unit  is  one  hundredth  of  the  rate  per  cent.,  con- 
versely the  rate  per  cent,  is  found  by  multiplying  the  rate 
per  unit  by  100. 

KuLE, —  Divide  the  percentage  by  the  basis.  Tlie  quo- 
tient will  be  the  rate  per  writ.  Midtiplij  the  rate  per  unit 
by  100.     The  product  ivill  be  the  rate  per  cent. 

Ex.  22.  What'per  cent,  of  700  is  70  ?    Ans.  10  per  cent. 

23.  What  per  cent,  of  375  is  125!;'  Ans.  33}  per  cent. 

24.  What  per  cent,  of  1000  is  125  ? 

25.  What  per  cent,  of  550  is  110?  Ans.  20  per  cent. 

182 


115.00 

690 

690 

.16-1 

4600   100 

4140 

16|- 

I>EaC£NTACJE.  §239 


26.  What  per  cent,  of  S675  is  $:^37.59  ?    Ans.  50  per  cent. 

27.  What  per  cent,  of  SIOOO  is  ^875  ? 

28.  What  per  cent,  of  $5000  is  ^250  ?  Ans.  5  per  cent. 

29.  What  per  cent,  of  $10000  is  S50  ?  Ans.  I  per  cent. 

30.  What  per  cent,  of  $150  is  $300  ? 

31.  What  per  cent,  of  3  JOO  is  4000  ?     Ans.  133a-  per  cent. 

32.  What  per  cent,  of  275  is  302.5  ?        x\.ns.  110  per  cent, 

33.  What  per  cent,  of  245  is  735  ? 

34.  What  per  cent,  of  200  is  500?  Ans.  250  per  cent. 

35.  What  per  cent,  of  325  is  2925  ?         Ans.  900  per  cent. 

36.  What  per  cent,  of  81.25  is  $1.50  ? 

37.  What  per  cent,  of  $1.00  is  $.375  ?     Ans.  37^  per  cent. 

38.  What  per  cent,  of  $.875  is  $.50  ?       Ans.  57^  per  cent. 
S9.  AVhat  per  cent,  of  $.66f  is  $.22|  ? 

40.  What  per  cent,  of  $.125  is  $.0625  ?     Ans.  50  per  cent. 

41.  A  commission  merchant  purchases  articles  amounting 

to  $247.75  ;  what  is  his  commission,  at  2^-  per  cent.? 

42.  What  is  the  commission  on  $312,  at  12  per  cent.? 

43.  A  merchant  insured   a   vessel    and    cargo,   valued    at 

$75000,  at  7f  per  cent. ;  what  did  he  pay  ? 

44.  What  premium  must  I  pay  for  the  insurance   of  hlj 

lifo5*the  policy  being  $5000,  and  the  rate  2.o5  per 
cent.?  Ans.  $117.50. 

45.  What  is  the  premium  for  insuring  $9450,  atf  per  cent.? 

46.  What  is  the  insurance  on  a  dwelling  and  furniture  val- 

ued at  $25550,  at  1,}  per  cent.?  Ans   $319,375-. 

47.  What  is  the  duty,  at  40  per   cent.,   on  French  broafll- 

cloths  valued  at  $15375  :  Ans.  $6150. 

48.  What  is  the  duty,  at  20  per  cent.,  on  $6250  worth   of 

Italian  silk  ? 
4.9.  At  7,}  per  cent.,  what  is  the  duty  on  an  invoice  of  Ge- 
neva watches,  valued  at  $7475?  Ans.  $560,635. 

183 


§239  PROPORTION. 


50.  At  50  per  cent.,  what  is  the  duty  on  a  ease  of  Leghorn 
hats,  worth  $1500  ?  Ans.  S750. 

51.  What  tax  should  be  paid  on  S17725  worth  of  real  es- 
tate, at  i  per  cent.? 

52.  What  is  the  tax  on  $261000,  at  .15  per  cent.? 

Anp.  $391.50. 

53.  What  is  the  tax  on  $17150,  at  60ct.  on  $100? 

Ans.  $102.90. 

54.  What  is  the  amount  of  a  dividend  of  3  per  cent.,   on 

$4200  of  bank  stock  ? 

55.  The  North  Carolina  Bailroad  company  declared  a  div- 
idend of  2|-  per  cent.  :  what  did  I  receive  on  14  shares 
of  $100  each  ?  Ans.  $35. 

56.  A  merchant  bought  broadcloth  at  $3.50  per  yard;  at 
what  price  must  he  sell  it,  to  gain  40  per  cent.? 

Ans.  $4.90. 
The  percentage  must  he  added  to  the  basis. 

57.  A  grocer  bought  candles  at  25ct.  per  lb.:  how  must  he 
sell  them,  to  gain  30  per'  cent.? 

58.  If  broadcloth  cost  $4.00  per  yd.,  how  much  will  it  bring 

at  a  loss  of  35  per  cent.?  Ans.  $2.60. 

The  percentage  must  he  suhtracted  from  the  basis. 

59.  A  dealer  bought  50bbl.  of  flour  at  $12  per  bbl.,  but.was 
forced  to  sell  it  at  a  decline  of  20  per  cent. :  what  did 
he  get  for  it  all  ?  Ans.  $480. 

60.  A  speculator  bought  $35000  worth  of  cotton,  and  sold 
it  at  a  loss  of  15  per  cent.  :  what  did  he  receive  for  it  ? 

61.  A  man  pays  $406.25  for  the  insurance  of  his  dwelling, 
valued  at  $32500  :  what  is  the  rate  per  cent.? 

Ans.  1-^:  per  cent. 

62.  A  vessel  worth  -SI 5400  was  insured  for  $539  ;  what  was 

the  rate  per  cent.?  Ans.  3-^-  per  cent. 

184 


PERCENTAGE.  §239 


63.  At  what  rate  per  cent,  will  the  insurance  on  $11500 

cost  $172.50  ? 

64.  A  man  bad  his  life  insured  for  $277.50':    what  was  the 
rate  per  cent.,  the  policy  being  $10000  ? 

Ans.  2.775  per  cent. 

65.  If  the  duty  on  ^3457  worth  of  goods  is  $1037.10,  what 

is  the  rate  per  cent.?  Ans.  oO  per  cent. 

66.  What  is  the  rate  of  duty,  when  $12657  worth  of  cloth- 
ing pays  $6828.50  ? 

67.  If  $15000  worth  of  property  pays  a  tax  of  $229.50,  what 
is  the  tax  on  $100  ?  Ans.  $  .51. 

66.  I  paid  a  hroker  $21,125  for  investing  $8450  in  Govern- 
ment stocks;  what  was  his  rate  of  brokerage? 

Ans.  \:  per  cent. 

69.  My  attorney  charged  me  $260.73f  for  collecting  $3476. 

50  :  what  was  his  rate  of  commission  ? 

70.  I  bought  a  farm  for  $4000,  and  sold  it  for  $5000  ;  what 

did  I  gain  per  cent.?  Ans.  25  per  cent. 

The  Jirst  cost  subtracted  /roin  the  selling  price  leaves  the 
total  gain. 

71.  I  bought  a  farm  for  $5000,  and  sold  it  for  $4000  ;  what 

did  I  lose  per  cent.?  Ans.  20  per  cent. 

The  selling  'price  subtracted  from  the  jirst  cost  leaves  the 

total  loss.      Observe  that  in  each  case  the  first  cost  is  the  basis. 

Hence  the  difference  in  the  ansiucrs  of  the  last  two  questions. 

72.  If  I  buy  calico  at  lOct.,  and  sell  it  at  12^-ct.,  what  do  I 
gain  per  cent.? 

73.  If  I  buy  calico  at  12^ct.,  and  sell  it  at  15ct.,  what  do  I 
gain  per  cent.?  Ans.  20  per  cent. 

74.  A  man  bought  a  house  for  $7625,  and  sold  it  for  $8387. 
50;  what  did  he  gain  per  cent.?         x\ns.  10  per  cent. 

1S5 


5; 240  PR  ^PORTION. 


75.   A  man  having  paid  S7625  for  his  house,  was  compelled 
to  !^eH  it  for  $6862.50  :  how  much  per  cent,  did  he  lose  ? 


70.  By  selling  an  article  for  ^1300,  I  gain  30  per  cent,  on 
it :  what  did  it  cost  me  ? 

$1300.00j  1.30  ^,240.  Model.— Divide  the  sell- 

■^^^        iSlOOO       ing  price  by  1  +  the  rate  per  unit. 
OUOU  (§  165.)  .  The  quotient  §1000  is  the 

first  cost. 

Explanation. — It  is  evident  that  l  +  thc  gain  per  unit 
:  1  ::  the  first  cost  +  the  whole  gain  :  the  first  cost.  But 
the  selling  price,  §1800j  is  evidently  the  first  cost  +  the 
whole  gain.  Then  since  the  second  term  of  the  proportion 
is  always  1,  it  is  easy  to  see  the  truth  of  the 

Rule. —  To  find  the  first  cost,  lohen  the  selling  price  and 
the  rate  'per  cent,  o/ gain  are  given.  Divide  the  selling  price 
hy  1  -j-  the  rate  per  unit.      The  quotient  loill  be  the  first  cost. 

Ex.  77.  By  selling  a  piece  of  muslin  for  ^50,  I  gain  100 
per  cent. ;  what  did  I  give  for  it  ?  Ans.  $25. 

78.  What  did  I  pay  for  eggs,  if  I  gain  33^^  per  cent,  by  sell- 

ing them  for  IGct.  per  doz.? 

79.  A  grocer  sold  a  lot  of  sugar  for  ^1058,  gaining  thereby 
15  per  cent. :  what  did  the  sugar  cost  him  ?     A.  §920. 

80.  A  merchant  sells  some  flour  for  $924,  and  gains  12  per 
cent,  on  it :  what  did  he  pay  for  it?  Ans.  $825. 

81.  The  selling  price  is  S1800,  the  gain  20  per  cent.:  what 
is  the  first  cost  ? 


82.  A  merchant  sold  a  quantity  of  cloth  for  S1410,  and  thus 
sustained  a  loss  of  6  per  cent. :  what  did  the  cloth  cost 
him  ? 

186 


PERCENTAGE.  §'241 


94        '^'50(3         §241.  Model. — Divide  the  selling 
470  price  by  1  — the  rate  per  unit.  (§165.) 

^>7Q  The  quotient  $1500  is  the  first  cost. 

~"~m) 

Explanation. — Evidently,  1  —  the  loss  per  unit  :  1  :: 

tiic  first  cost  — the  whole  loss  :  the  first  cost.     But  the  first 

cost  — the  whole  loss  is  evidently  the  selling  price,  ^1410. 

The  second  term  of  this  proportion  is  always  1,  and  hence 

the  following 

EuLE. —  To  Jin  J  the  Ji  rat  coiif,  when  the  sellhig  price  and 

the.  rrtfe  j)er  cent,  of  loss  are  given.      Divide  the  selling  pric^ 

hj/  \—thc  rate  per  unit.      The  quotient  will  he  the  fir.st  co?t. 

Ex.  83.  The  selling  price  is  ?$8000  ;  the  loss  20  percent.: 

what  is  the  first  cost  ?  Ans.  $10000. 

'<4.  By  selling  flour  at  $12.25  per  bbl.,I  lost  12^  per  cent, 
of  wliat  it  cost  me  ;  what  did  it  cost? 

ST).   I  remitted  ^3150  to  my  commission  merchants  to  lay 
out  in  groceries  after  retaining  5  per  cent,  of  what  he 
spent:  how  much  did  he  spend  for  me?      Ans.  ^3000. 
This  is  precisely  similar  in  principle  to  the  foregoing. 

86.  How  much  sugar  at  lOct.  per  lb.  can  I  get  by  remit- 
ting $864,871-  to  a  merchant  who  cliarges  5  per  cent. 
commission  1'  Ans.  34751b. 

S7.  How  much  stock  at  ^5  per  cent,  advance  can  I  buy  for 
S1265  ? 

.S8.  How  much  stock  at  15  per  cent,  below  par  can  I  buy 
farC935?  Ans.  §1100. 

.Si).  A  father  settled  his  sou  with  property  worth  SIOOOO  : 
the  first  year  he  lost  20  per  cent,  of  it,  and  the  second 
year  he  gained  25  per  cent,  of  what  he  had  left ;  how 
much  had  he  then  't  Ans.  ^lOOOO. 

187 


^242  PB.OPORTION — PERCENTAGE. 

90.  A  merchant  sold   some  sugar  for  31402.50,  and  lost 
thereby  15  per  cent.     Wh.it  did  it  cost  him? 

91.  How  much  stock  at  :i  discount  of  3^-  per  cent,   caii  I. 
bought  for  f-5790  ?  Aus.  S6000. 

SIMPLE  INTEREST. 

§242.  Interest  is  the  prict  paid  bj  the  borrower  for 
the  use  of  money  loaned. 

§  243.  The  sum  of  money  on  which  interest  is  calculated 
is  called  the  principal. 

§  244.  The  sum  of  the  principal  and  interest  is  called  the 
amount. 

§245.  The  price  paid  for  the  use  of  one  hundred  doUarpi 
one  year  is  called  the  rale  per  cent,  per  annum. 

Ex.  1.  What  is  the  interest  of  ^875,  for  2yr.  lOmo.  20da.. 

at  7  per  cent,  per  annum  ? 

^375         2yr.  lOmo.  20da.  7n.  c. 
.07  '  §  246.  Model.— 

i26^=lyr.                                          Multiply^ the  priu. 
2  ......  iU.wK.. 


cinai  by  tlie  rale  per 

unit.  (§!«3.)     The 

^52.50=2yr.  product,  S26.25,  is 

13.125=:6mo. -j  tj,e    interest   for    i 

6.562=3mo.   -lOmo.  y^,,^^    Multiply  the 

2.187= Imo.  j  interest  for  lyr.  by 

•^■o^']"'^^^-  \  20da.  2.     The  product  i^ 

.364=   5da..  |  !$52.50,  the  interest 


$75.831  =  2yr.  lOmo.  20dn.  for  2yr.    6mo.  is  one 

half  of  lyr.  Divide 
the  interest  for  lyr.  by  2.  The  quotient  is  $13,125,  the 
interest  for  6mo.  3mo.  is  one  half  of  6mo.  Divide  the  in- 
terest for  6mo.  by  2.     The  quotient  li  36,562,  the  iuterest 

188 


SIMPLE   INTEREST.  §246 


for  oiiiO.  Imo.  is  one  third  of  3ino.  Divide  the  iuterest 
for  Smo.  by  3.  The  quotient  is  82.187,  the  interest  for  1 
mo.  loda.  is  one  half  of  Imo.  Divide  the  interest  for  1 
mo.  by  2.  The  quotient  is  SI. 098,  the  interest  f\;r  15da. 
5da.  is  one  third  of  15da.  Divide  the  iuterest  for  15da.  by 
o.  The  quotient  is  S  .oG4,  the  inleresi  for  5da.  Add  the 
partial  interests  together.  The  sum  is  S75.831,  the  inter- 
im est  for  the  whole  time. 

ExPLA-NATiOM. — Since  the  rate  is  7  percent,  per  annum, 
the  interest  of  the  given  principal  for  1  year  is  found  by 
multiplying  the  principal  by  the  rate  per  unit.  Thus  far 
the  work  is  simple  percentage.  Far  longer  or  shorter  pe- 
riods of  time  the  interest  is  proportional  to  the  time  :  hence 
we  take  such  aliquot  parts  of  the  interest  for  1  year,  &c., 
as  the  periods  in  question  severally  require. 

In  the  calculation  of  interest,  a  month  is  considered  equal 

■  30  days,  and  a  year  to  360  days. , 

Rule. — Multij^ly  the  principal  hi/  the  rate  per  unit.  The 
product  will  he  the  inf.erefi.t  for  1  year. 

Alidtiplij  the  'interest  for  1  year  hy   ihe  number  of  years , 

and  take  aliquot  parts  for  periods  of  time  less  than  a  year. 

To  ftriii  the  amount,  add  the  interest  to  the  principal. 

Ex.  2.  What  is  the  interest  of  850  for  2yr.  at  6  per  cent. 

per  annum  ?  Ans.  ^G. 

3.  What  is  the  interest  of  S75  for  6mo.  at  7  per  cent,  per 

annum  't 

•i.  What  is  the  amount  of  SlOO  for  9mo.  at  8  per  cent,  peu 

annum  ?  Ans.  ^106. 

5.  What  is  the  interest  of  3125  for  3yr.  at  9  per  cent,  per 

annum '/  Ans.  $33.75. 

G.  What  is  the  iuterest  of  %22b  for   2yr.  Gmo.   at  10  per 

cent,  per  annum  ? 

189 


^24G  riioroiiTiON — rELCEM..^ui:. 


7.  What  IS  tlie  iiitere.st  of  ^150!75  for  lyr.  omo.  at  5  per 
cent,  per  annum  't  Ans.  $10.67. 

8.  What  is  the  amount  of  3176.50  for  2yr.  9mo.  at  G  per 

cent,  per  annum  ?  Ans.  S204.45. 

9.  What  is  the  interest  of  s^305.50  for  %r.  5mo.  15<3a.  at 

6  per  cent,  per  :iiinnni? 

10.  What  is  the  interest  of  S574.05  for  -^jr.  7iiio.  25da.  at 
5  per  cent,  per  annum?  Ans,  81oS.75. 

11.  What  is  the  intere^.t  of  8615.49  for  fiyr.  llmo.  22da.  at 

7  per  cent,  per  annum  ?  Ans.  $257.47. 

12.  Find  the  amount  of  ?-777.75   for   !>yr.  2mo.  20da.  at  8 
per  cent.,  per  annum. 

13.  Find  the  interest  of  ^'1225  for  ojr.  5n)0.  oda.  at  5  per 
cent,  per  annum.  Ans.  ?332.G2. 

14.  Find  the  intere.s:  of  S1525.25  for  lyr.  2mo.   at  8   per 

cent,  per  annum.  Ans.  S  142.356. 

15.  Find  the  interest  of  62790  for  23'r.  7mo.  at  9  per  ccrit. 

per  annum. 

16.  Find  the  amount  of  $1724.25  for  12yr.  Omo.   at   8  per 

cent,  per  annum.  Ans.  ^'3448.50. 

17.  Find  the  interest  of  $3500  for  7yr.  omo.  IGda.  at  7  per 

cent,  per  annum.  Ans.  81783. 0^'5. 

18.  Find  the  interest  of  -^4275  for  16yr.  Smo.  at  6  per  cent, 

per  annum. 

19.  Find  the  interest  of  S5550  for  15yi'.  llmo.  27da.   at  0 

per  cent,  per  annum.  Ana.  v?79S7.83{,. 

20.  Find  the  amount  of  $2c;95  for  tS)  r.  4mo.  at  12  per  cent. 
per  annum.  x\ns.  $5990. 

21.  Find  the  interest  of  $3827  for  17yr.  3mo.  15da.  at  10 

per  cent,  per  annum.  Ans.  $6617.52. 

190 


SIMPLE   INTi-UFST.  §247 

CONCISE  METHOD  FOR  6  PER  CJ^NT.  PER  AKNU.\L 


Ex.  22.  What  is  the  interest  of  SMT.OO  fur  2yr.  Cmc^  IS 
da.  at  G  per  cent,  per  annum  ? 

2yr.  6mo.  18da.  =  30.Grao.  {^  247.    M(*DKr .— Rcdnre 

200)30.6         ^247.50  ^^^  ^'^^"  ^^"^^  ^''  nsonth^,. 

~YFb                 1  ^S  (§1^^.)    Divide  tho  number 

*                   h.'okI  «^^  months  by  20U.    (^  165. ) 

lOQ^'^A  Multiply   iho   principal    by 

l^d/^0  this  quotient,  (i^  162.)    The 

J_Z^iL  product  is  ?P>7.8(1A,  the  in- 

$37.86750  terest  required. 

Explanation. — Since  the  rate  is  6  per  cent,  per  annum, 
or  for  12  mouths,  one  half  of  the  number  of  montlis  is  the 
rate  per  cent,  for  any  length  of  time  :  and  this  rate  per  cent, 
divided  hy  100,  gives  the  corresponding  rat(3  per  unit,  by 
which  the  principal  must  be  multiplied,  to  find  the  interest. 

For  any  other  rate,  we  may  find  the  interest  at  G  per 
cent.,  and  increase  or  diminish  it,  as  the  case  may  require. 
For  instance,  for  7  per  cent.,  add  to  the  interest  found  by 
this  method,  one  sixth  of  its?lf :  for  5  per  cent.,  from  the 
interest  thus  found  subtract  one  sixth  of  itself;  &c.  Or, 
generally,  find  the  interest  at  6  per  cent.,  divide  it  by  6, 
and  multiply  the  quotient  by  the  given  rate. 

Rule. — Divide  the  number  of  monthly  in  the  given  time 
hy  200,  and  multiplij  the  jprincipal  hy  the  quotient.  The 
product  ivill  he  the  interest. 

Or,  3Iidtiply  the  number  of  years  hy  6,  and  divide  the 
product  hy  100  ;  Divide  the  number  of  months  hy  2,  and 
divide  the  quotient  hy  100  .-  Divide  the  number  of  days  by 

191 


§248  PROPOllTION — PERCENTAGE. 

6,  and  diclile  (he  quotient  hy  1000  ;   Add  these  three   results 
fo<jctherj  and  multiply  the  principal  hy  their  sum. 

After  Jill  din  f/  the  interest  at  6  per  cent.,  as  above,  to  find 
the  interest  at  any  other  rate  ;  Divide  the  interest  at  6  per 
rent,  hy  G,  and  multiply  the  quotient  hy  the  required  rate. 


SECOND  METHOD  FOR  6  PER   CENT. 


Ex.  23.  Find  the  interest  of  8275.75  for  Syr.  lOmo.  21 
(la.  at  6  per  cent,  per  annum. 

Syr.  lCmo.=46mo.  §248,    Model. — Reduce   the 

3)21         v^275.75  years  and  months  to  months. — 

— 7               .467  Divide  the  number  of  days  by 

T  AOAoT  3.     Annex  the  quotient  to  the 

Ifi'S^^O  number  01  months.    Divide  this 

^^,^oAa  icsult  bv  1000.     Multiply  the 

principal  by  this  quotient.    Di- 

2)812877525  ^.iae  this  product  by  2.     The 

$64.3876  quotient  is  sj61.38£,  the  inter-. 
est  required. 

Explanation, — This  method  is  evidently  the  same  in 
principle  as  the  preceding,  and  is  preferable  to  the  other 
only  on  account  of  its  greater  freedom  from  liability  to 
fractions.  Of  course,  the  multiplier  iu  each  of  these  lueth- 
•ods  must  be  considered  abstract. 

Rule. —  To  th6  number  of  months  annex  one  third  of  the 
humber  of  day^s.  Divide  the  number  thus  produced  by  1000. 
Multiply  one  half  of  the  principal  hy  this  quotient. 

Or,  Multiply  the  whole  principal  by  this  quotient,  and 
divide  the  product  by  2. 

The  interest  for  any  other  rate  may  be  found  as  in  §  247. 

192 


SIMPLE    INTEREST.  §249 


CONCISE  METHOD  FOR  A^Y  S1.4TE  PER  CENT. 


Ex.  24.  Find  the  interest  of  ^^330. 60  for  6mo.  15da.  at  S 

per  cent,  per  annum. 

12,00)^3,60.60 

"~$  .3005 

g  5  §  249.  jModel. — Divide  the  princi- 
pal by  1200.  Multiply  this  quotient 
by  6.5.  M'lUiply  this  product  by  8. 
The  product  h  $15.62|-,  the  required 


15025 

18030 


^l,95b25     interest. 


^■15.62600 

ExPLVNATLON. — The  principal -^100=:the  interest  for  1 
year  at  1  per  cent.  This  interest-^- I2=the  interest  for  1 
month  at  1  per  cent.  This  last  x  6. 5  =  the  interest  for  6.5 
months  at  1  percent.  And  thtsx8  =  the  interest  for  6.5 
months  at  8  per  cent. 

Rule. — Divide  the  principal  by  1200.  Multiply  the  quo- 
tient by  the  wumber  of  months  in  fh'i  given  time,  and  this 
product  by  the  rate  per  cent.  This  last  product  will  be  the 
interest.- 

Either  of  the  above  methods  may  be  used  in  any  case. 

*Ex.  2"'.  Find  the  interest  of  i?l 3 19.50  for  9  days,  at  6 
per  cent.  |rr  annum. 

26.  Find  ilie  interest  of  $36")8.75  for  17  days  at  6  per  cent. 

per  nnaum.  "  Int.  SI 0.366. 

27.  Find  the  interest  of  $5739.2,3  for  2mo.  24da.  at  6  per 

cent,  per  annum.  Int.  $80,349. 

28.  Find  the  amount  of  $3738^.375  for  2mo.  6da.  at  6  per 

cent,  per  annum.  Amt.  $38096.88. 

M  193 


§249      ,     PROPORTION — PERCENTAGE. 


29.  Find  the  amount  of  ^1665.25  for  lyr.  llmo.  9da.  at  C 
per  cent,  per  annum.  Amt.  S1859.25. 

SO.  Find  the  interest  of  ^4336.30  for  4yr.  8mo.  13da.  at  6 
per  cent,  per  annum. 

?>1.'  Find  the  interest  of  ^2758.50  from  July   3,  1846,  to 

-May  19;  1855,  at  6  per  cent,  per  annum. 

Int.  $i469.36. 

To  find  the  interval  of  time,  the  earliest  date  must  be 

-iQrr        r         to  J  subtractcd  from  the  latcst.    In 

Ibooyr.  omo,  lyda, 

1846  "    7  "       3  "   •        ^^^^  subtraction,  the  number 
^  u  10  «'     IQ  <'  ^^  emc]!  month  in  the  calendar 

is   used,    and    each  month  is 
toaken  as  equal  to  30  days. 

Ex.  32.  Find  the  amount  of  $8140.75  from  Dec.  9,  1847, 
to  Apr.  27,  1855,  at  6  per  cent,  per  annum. 

Amt.  §11747.10. 

33.  Find  the  interest  of  $3^219.15,  from  Apr.  8,  1850,  to 
June  15,  1855,  at  7  per  cent,  per  annum. 

S4.  Find  the  interest  of  $6813.45  from  Mar.  5,  1855,  to 
Oct.  8,  1862,  at  8  per  cent,  per  annum. 

Int.  $4138.035. 

35.  Find  the  interest  of  $856.85  for  6yr.  8mo.  9da.  at  8  per 

cent,  per  annum.  Int.  $458,699. 

36.  Find  the  amount  6f  $742.40  from  June  24,  1854,  to 

Mar.  13,  1860,  at  7  per  cent,  per  annum. 

37.  Find  the  interest  of  $171.80  from  July  29,   1857,  to 

Sept.  1,  1861,  at  10  per  cent,  per  annum.    Int.  $70.24. 

38.  Find  the  interest  of  $670.70  from  Apr.  7, 1859,  to  Oct. 

13,  1862,  at  9  per  cent,  per  annum. 

Int.  $212,276. 

^9.  Find  the  interest  of  $976.18  from  Mar.  1, 1861,  to  Feb. 
10,  1862,  at  8i  per  cent,  per  annum. 

194 


SIMPLE   INTEREST.  §249 


40.  Find  the  interest  of  ^375.85  from  Jan.  19,  1860,  to 

Jan.  1,  18G2,  at  11  per  cent,  per  annum. 

Int.  $80,619. 

41.  Find  the  amount  of  $6.89  from  June  11, 1860,  to  June 

1,  1862,  at  9  per  cent,  per  annum.  Amt.  ^8.11. 

42.  What  is  the  interest  of  S89.96  for  2yr.  3mo.  16da.at  8 
per  cent,  per  annum  ? 

43.  What  is  the  interest  of  $325  for  6jr.  7mo.  27da.  at  7-^ 
per  cent,  per  annum  ?  Ans.  $156.88. 

44.  What  is  the  amount  of  $1728  from  Dec.  29,  1859,  to 

Oct.  9,  1852,  at  10  per  cent,  per  annum  ?  Ans.  152208. 

45.  W^hat  is  the  interest  of  $160.08  from  May  1,   1851,  to 
Sept.  9,  1854,  at  7  per  cent,  per  annum? 

46.  What  is  the  interest  of  $18.62  for  3yr.  18da.  at  5  per 
cent,  per  annum  ?  Ans.  $2,839. 

47.  What  is  the  interest  of  i£17,  6s.  9d.  for  18mo.  at  6  per 
cent,  per  annum  ? 

£17,  6s.  9d.=£17.3375         The  principal  must  first  be 

^^     reduced  to  pounds,  and  then 

^1.560375     the  interest  may  be  found  by 
c£l. 56=^1,  lis.  2^d.         any   one   of  the    preceding 

methods. 
]i]x.  48.  What  is  the  interest  of  ^6427, 18s.  9d.  for  2  years 
at  5:}  per  cent,  per  annum  ? 

49.  What  is  the  amount  of  ,£1096,  15s.  6d.  for  4  years  at 

6-1-  per  cent,  per  annum  ?         ^     Ans.  ^61381,  18s.  8d. 

50.  What  is  the  amount  of  £120,  10s.  for  2yr.  6mo.  at  4a 
per  cent,  per  annum  ?  Ans.  j£134,  36s.  l|,d. 

51.  What  is  the  interest  of  £270,  10s.  >0d.  for  lyr.  4mo.  20 
da.  at  7  per  cent,  per  annum  ? 

52.  What  is  the  interest  of  1775fr.  75cent.  for  3yr.  6mo.  at 

6  per  cent,  per  annum  1  Ans.  372fr.  90cent. 

195 


§250  PROPORTION — PERCENTAGE. 

53.  What  is  the  interest  of  2070fr.   65cent.  for  2yr.  8mo. 
20da.  at  7  per  cent,  per  annum  ?      Ans.  394fr.  STcent. 

54.  What  is  the  amount  of  o29Tfr.  15cent.  for  oyr.   15da. 
at  8  per  cent,  per  annum  ? 

55.  What  is  the  interest  of  10720fr.  25cent.  for   5yr.   7mo. 

lOda.  at  5  per  cent,  per  annum  ?  Ans.  3007. G2fr. 

56.  What  is  the  amount  of  20625fr.  SOcent.  for   (5yr.    6mo. 

6da.  at  6  per  cent,  per  annum  1  Ans.  28689. 79fr, 


PARTIAL  PAYMENTS. 


The  method  here  given  is  the  one  enjoined  by  the  Su- 
preme Court  of  North  Carolina;,  and  used  iu  most,  if  not 
all,  the  States  of  the  Confederacy. 

§  250.  Rule. —  Find  the  amount  of  tlie  given  principal  to 
tJie  titne  of  the  first  p)ciymtnt^  arid  if  this  pa 7/ men  f  is  greater 
than  the  interest  then  due,  subtract  the  payment  from  the  a- 
mount.  Consider  the  remainder  as  a  second  pr in cipaly  and 
find  the  amount  of  it  from  the  time  of  the  first  payment  to 
the  time  of  the  second,  and  if  the  second  payment  is  greater 
fhan  the  interest  last  found,  subtract  the  second'  payment 
from  the  second  amount,  and  consider  the  remainder  as  a 
third  principal:  and  so  on. 

But  if  any  payment  is  less  than  its  corresponding  inter- 
est, find  the  amount  of  the  same  principal  to  the  time  of  tJie 
next  payment,  and  if  the  sum  of  these  two  payments  is 
greater  than  the  interest  then  due,  subtract  their  sum  from 
the  amount :  but  if  the  sum  (fi  the  two  payments  is  less  than 
the  interest  then  due,  extend  the  time  until  the  sum  of  the 
payments  made  shall  exceed  the  interest  due  at  the  time  of 

the  last  payment. 

196 


PARTIAL  PAYMENTS.  §250 


The  principle  of  tlie  rule  is  that  the  payment  of  a  part 
of  the  debt  shall  not  increase  the  debt. 

Ex.  57.  $725.50.  Richmot^d,  Va.,  Jan.  1,  1858. 

One  day  after  date,  I  promise  to  pay  J.- Jones,  or  order, 
seven  hundred  and  twenty-five  dollars  and  fifty  cents,  for 
value  received.  ^  (^../^  (g^ 

On  this  note  were  the  following  endorsements  : 
Mar.  16,  1858,  $100.00 

May    16,  1859,  25.50 

.     July     1,  1861,  300.00 

How  much  was  due  Oct.  8,  1862  ? 

SOLUTION. 

Original  Principal,  $725.50 

Interest  to  Mar.  16,  1858,— 2ra.  15da.,  9.068 

Amount  then  due,  ^734.568 

Amount  then  paid,  100. 

Second  Principal,  $634,568 
Interest  from  Mar.  16,  1858,  to  May  16,  1859, 

$44,419 
Amount  then  paid  (less  than  interest)      25.50 
Interest  from  Mar.  16,  1858,  to  July  1,  1861, — 

3y.  3m.  15d.  125.327 

Amount  then  due,  $759,895 

Sum  of  the  two  payments,  325.50  ^ 

Third  Principal,  $434,395 

Interest  from  July  1,  1861,  to  Oct.  8,  1862,— 

ly.'3m.  7d.  33.086 

Amount  due  Oct.  8,  1862,  $467,481 

197 


§250  PROPORTION PERCENTAGE. 


')8.  $3256.37.  Lincolnton,  N.  C,  Mar.  12,  1853. 

On  demand  I  promise  to  pay  to  the  order  of  J.  Rein- 
liardtj  three  thousand  two  hundred  and  fifty-six  dollars  and 
thirty-seven  cents,  for  value  received. 

On  this  note  were  the  following  endorsements  : 
July  12,  1855,       received  $654.33 
Sept.  20,  1857,  "         $246.50 

Jan.     5,  1859,  "         $945.87 

What  was  the  "balance  due  Sept.  7, 1860  ?  Ans.  $2755.41. 


59.  $108.43.  Columbia,  S.  C,  Dec.  9,  1857. 
With  interest  from  date,  for  value  received,  I  promise  to 

pay  J.  Townsend  or  order  oiic  hundred  and  eight  dollars 
and  forty-three  cents.  ^^    C^^^/.     ("^^ 

Endorsements.  Mar.  3,  1858,  received  $50.04  ;  Dec.  10, 
1858,113.19;  May  1,  1860,  S?50.11.  How  much  was  due 
Oct.  9,  1862  ?  Ans.  1^55.844 

60.  A  note  was  given  at  Savannah,  Geo.,  Apr,  16,  1856, 
for  $450.  On  it  the  following  endorsements  were 
made  : — Jan.  1,  1857,  received  ^20  ;  Apr.  1,  1857, 
$14;  July  16,  1857,  %Z\  ;  Dec.  25,  1857,  ^10  ;^  July 
d:,  1858,  $18.     What  balance  was  due  June    1,   1859  ^ 

Note. — When  r.o  rate  of  interest  is  mentioned  in  a  note,  tiie 
legal  rate  at  the  place  where  it  is  given  is  to  be  used.  In  Louirfiana 
the  legal  rate  is  5  per  cent.  :  in  Arkansas,  Kentucky,  Maryland, 
Mii-souri,  North  Carolina,  Tennessee,  and  Virginia,  it  is  6  per  cent.: 
in  South  Carolina  it  is  7  per  cent.  ;  and  in  Alabama,  Florida,  Geor- 
gia, Mississippi,  and  Texas,  it  is  8  per  cent. 

198 


COMPOUND   INTEREST.  |25l 


COMPOUND  INTEREST. 


§  251.  Compound  Interest  is  the  interest  on  both  princi- 
pal and  interest  when  the  interest  is  not  paid  as  it  falls  due. 
In  ordinary  business  transactions  it  is  not  allowed  by  law; 
but  in  a  few  classes  of  debts  it  is  required  that  the  inter- 
est shall  be  compounded  annually.  In  such  cases,  the  in- 
terest for  one  year  is  added  to  the  principal;  this  amount 
becomes  the  principal  for  the  second  year ;  its  amount  for 
the  third  year,  and  so  on  to  the  last  year  or  part  of  a  year. 
The  original  principal  subtracted  from  the  final  amount 
gives  the  compound  interest. 

Ex.  Gl.  What  is  the  'compound  interest  of  $525.75  for 
3yr,  Gmo.  at  6  per  cent.,  interest  due  annually  ? 

SOLUTION. 

Original  Principal,  $525  75 

Interest  for  the  first  year,  .  31.545 

Amount, — Second  Principal,  $557,295 

Interest  on  $557,295  for  the  second  y^car,  33.437 

Amount, — Third  Principal,  $590.73^ 

•    Interest  on  $590,732  for  the  third  year,  35.443> 

Amount,— Fourth  Principal,  .$6267176 

Interest  on  $626,175  for  the  remaining  6mo.,  18.785 

Total  Amount  at  Compound  Interest,  $644,960 

Original  Principal,  525.75 

Compound  Interest,  $119.21 

Ex.  62.  What  is  the   amount   at  compound  interest  of 
$500  at  6  per  cent,  for  4yr.  3ino.,  interest  due  annually  ? 
63.  What  is  the  amount  of  $1000  for  7  years  at  7  percent^, 
compounded  annually  ? 

199 


§252  PROPORTION — PERCENTAGE. 


64.  What  is  the  amonnt  of  $1000  for  6  years  at  6  per  cent., 
compounded  semi-annually?  Ans.  ^1425.76. 

^.  What  is  the  interest  of  $1000  for  4  years  at  6  per  cent., 
compounded  quarterly  ?  Ans.  1268.98. 


DISCOUNT. 


§  252.  Discount  is  a  deduction  made  for  the  payment  of 
money  before  it  is  due. 

§  253.  The  present  worth  of  a  future  debt  is  that  sum 
which,  at  ordinary  interest,  will  amount  to  the  debt  at  the 
time  it  becomes  due.  The  present  worth  bears  the  same 
relation  to  the  debt,  that  the  principal  h^d^rs,  to  the  amount. 

The  problem  to  be  solved,  then,  is,  having  given  the 
amount,  the  time,  and  the  rate,  to  find  the  principal  and 
the  interest. 

§  254.  Rule. — Find  the  amount  of  $1  for  the  given  time 
at  the  given  rate.  Then,  as  the  amount  of  $1  is  to  $1,  so 
is  the  amount  of  the  debt  to  its  present  worth. 

To  find  the  discount,  subtract  the  present  ivorth  from  t\e 
amount  of  the  debt.  Or  say,  as  the  amount  of  $1  is  to  its 
interest^  so  is  the  amonnt  of  the  debt  to  the  discount. 


Ex.  66.  What  is  the  present  worth,  and  what  is  the  dis- 
count, of  a  note  due  6  months  hence  for  $550  at  6  per  cent.? 

SOLUTION. 

Amount  of  $1  for  6  months  at  6  per  cent.,  $1.03. 

$1.03  :  $1  ::  $550  :  $533.98,  pres,ent  worth. 

$550  — $533.98=$16.02=the  discount. 

Or,  $1.03  :  $.03  ::  $550  :  $16.02,  the  discount. 

200 


DISCOUNT.  §255 


Ex.  67.  What  is  the  present  worth  of  a  note  for  $245, 
due  1  year  hence  when  the  rate  of  interest  is  6  per  cent.? 
68.  What  discount  should  be  allowed  on  a  note  for  $525,  if 

paid  3mo.  before  it  is  due,  interest  being  at  7  ner  cent,? 
C9.  What  is  the  present  worth  of  a  debt  of  $375.50,  due  in 

Tnio.  15da.,  if  interest  is  at  8  per  cent.? 

70.  What  is  the  discount  of  a  note  for  $725,  due  in  lOmo. 

lOda.,  interest  being  7  per  cent.? 

71.  In  Mobile,  Ala.,  one  man  gave  another  his  note  for 
$247. 50j  due  twelve  months  after  date.  What  was  the 
present  worth  of  the  note  ? 

72.  What  discount  would  be  allowed  at  New  Orleans  on  a 

debt  of  $650,  due  9  months  hence  ? 

73.  What  is  the  present  worth,  at  Little  Rock,  Ark.,  of  a 

note  for  $769.35,  due  5mo.  18da.  hence? 

74.  What  is  the  proper  discount  on  a  debt  of  $75.75,  due 

7mo.  hence  at  Memphis,  Tenn.? 

75.  What  is  the  present  worth  of  ^1250,  due  12  months 
hence  at  Galveston,  Texas  ? 

76.  What  is  the  discount  of  $250,  due  8mo.  hence  at  Lex- 
ington, Ky.? 

77.  What  is  the  present  worth  of  $55.55,  due  7mo.  benee 
at  St.  Louis,  Mo.? 


BANK  DISCOUNT. 


§  255.  The  present  worth  or  proceeds  of  a  note  payable  in 
bank  is  the  remainder  obtained  by  subtracting  from  its  face 
its  interest  for  the  time  it  has  to  run,  including  three  addi- 
tional days — called  days  of  grace. 

Thus,  if  I  deposit  with  the  Cashier  of  the  Bank  of  Cape 

201 


§25(5  PROPORTION — PERCENTAGE. 


Fear  my  note  for  $1000  due  in  60  chxys,  lie  will  pay  me  on 
it  only  $1000— the  interest  of  $1000  for  63  days,  that  is, 
$989.50. 

§256.  The  bank  discount  of  a  note  not  yet  due  is  the  in- 
terest of  the  face  of  the  note  for  three  days  more  than  the 
time  it  has  to  run. 


Ex.  78.  What  is  the  present  worth  in  bank  of  a  note  for 
$500  due  in  30  days,  at  6  per  cent.? 

SOLUTION. 

Face  of  the  note,  '      $500. 

nterest  of  $500  for  Soda., — hank  discount,  2.75' 

Present  Worth  or  proceeds, 

Ex.  79.  What  is  the  proceeds  of  a  note  due  in  bank   60 
da.  hence  for  $250  at  6  per  cent.? 

80.  What  is  the  bank  discount  on  a  note  for  $750   due  in 
bank  in  90  days,  at  6  per  cent.? 

81.  What  discount  would   a  Kank  require   on   a  note   lor 

$550.75,  due  90  days  hence  at  8  per  c(5nt.? 

82.  What  is  the  present  worth  of  a  note  due  in  bank  90da. 
hence  for  $333.33  at  6  per  cent.? 

83.  What  is  the  face  of  a  note  due  60da.  hence,  if  its  pres- 
ent worth  in  bank  is  $500,  interest  being  at  6  per  cent.? 

§257.   The  present  worth  of  $1  :  $1   ::  present  worth  of 
the  note  :  face  of  the  note.     In  this  case,  $.9895  :  $1    :: 
$500  :  the  answer. 

Ex.  84.   What    sum,  payable  in  90  days,  will   produce 
$750,  if  discounted  at  a  bank  at  6  per  cent.? 
85.  What  sum,  payable  in  60  days,  will  produce  $S000,  if 
discounted  at  bank  at  7  per  cent.? 

202 


DISCOUNT. 


§257 


86.  Far  what  amount  must  a  note  be  drawn,  payable  in  30 
days,  so  that,  if  discounted  in  bank  at  5  per  cent.,  the 
proceeds  will  be  $250  ? 

87.  What  must  be  the  face  of  a  note  payable  in  bank  in  90 

days,  so  that,  if  discounted  at  6  per  cent.,  its  present 
worth  may  be  $75.75  1 


Showing  the  number  of  days  from  any  day  of  one  month  to 
the  same  day  of  any  other  month  next  following. 


From  any 
(iay  of 

To  the  same  day  of  the  uoxt               j 

Jan. 

385 

Feb. 

31 

Mar. 
59 

Apr. 

May 

120 

June 

151 

July 
181 

Aug. 

212 

Sept. 

243 

Oct. 

273 

Nov. 

304 

Dec. 

334 

Jan. 

Feb. 

334 

36". 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

Mar. 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

Apr. 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244  1 

May 

245 

276 

304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

1  June 

^14 

245 

273 

304 

334 

365 

30 

61 

92 

122 

153 

183: 

July 

184 

215 

243 

274 

304 

335 

365 

31 

(12 

92 

123 

i53i 

Vug. 

153 

184 

212 

243 

273 

304 

365 

31 

61 

92 

122il 

:  Sept. 

122 

153 

181 

2111 

242 

303 

334 

365 

30 

61 

91;' 

Oct. 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61|i 

Nov. 

Gl 

92 

120 

151 

181 

212 

242 

273 

304 

334 

365 

30i 

Dec. 

M\ 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 

To  find  the  interval  of  time  between  Sept.  3,  1862,  and 
May  19,  1863.  Find  Sept.  in  the  left  hand  column  and 
May  in  the  upper  line  :  then  at  the  right  of  Sept.  and  un- 
der May,  is  242,  the  number  of  days  from  Sept.  3  to  May  3. 
To  this  add  16,  the  number  of  days  from  May  3  to  May  19. 
The  sum  258  is  the  number  of  days  required. 

Again,  from  Jan.  25,  to  Sept.  9,  is  (243-16)  227  da^s, 

203 


§258  PROMISCUOUS  PROBLEMS. 


PROMISCUOUS  PROBLEMS. 


1.  In  what  time  will  $100  amount  to  $200  at  6  per  cen 
simple  interest  ? 

§258.  As  the  interest  of  the  given  principal  for  1  year 
the  given  interest  ::  1  year  :  the  number  of  years. 

In  this  case,  as  $6  :  $100  ::  lyr.  :  16yr.  Smo.,  the  a' 
swer. 

2.  In  what  time  will  $200  gain  $50  interest  at  6  per  cent, 
per  annum  ? 

3.  In  what  time  will  $500  gain  $49  interest  at  7  per  cent.? 

4.  In  what  time  will  $1000  gain  $10  simple  interest  at  5 
per  cent,  per  annum  ? 

5.  At  what  rate  will  $100  gain  $15  interest  in  2yr.  6mo.? 
§  259.  As  the  interest  of  the   given  principal   at  1   per 

cent.  :  the  given  interest  ::  1  :  the  rate  per  cent. 
In  this  case,  as  $2.50  :  $15  ::  1  :  6j  the  answer. 

6.  At  what  rate  will  $250  gain  $250  interest  in  lOyr,? 

7.  At  what  rate  will  $427.25  gain  $143.60  in  Syr.  4mo. 

lOda.?  Ans.  10  per  cent. 

8.  At  what  rate  will  $746  gain  $83.92  in  2yr.  3mo.? 

9.  "What  principal  will  gain  $174.56  in  lyr.  7mo.  at  7  per 
cent.? 

§  260.  As  the  interest  of  $1  for  the  given  time  at  the 
given  rate  :  the  given  interest  ::   $1  :  the  principal. 

In  this  case,  $.11081  :  $174,56  ::  $1  :  $1575,  the  an- 
swer. 

10.  What  principal  will  gain   $42  in   Syr.  6mo.   at  6  per 
cent.?  Ans.  $200. 

11.  What  principal  will  gain  $210  in  Syr.  at  6  per  cent.t 

204 


PROMISCUOUS  PROBLEMS.  §260 


• 


12.  What  principal  will  gain  $400  in  4  years  at  8  per  cent.? 

13.  What  is  the  fourth  root  of  810000  ?  Ans.  30. 

14.  What  is  the  value  of  2.8.5.7.11  ? 

15.  What  are  the  prime  factors  of  1800  ? 

16.  A  commission  merchant  sold  goods  worth  $9072;  what 

was  his  commission  at  2^-  per  cent.?  Ans,  $220.80. 

17.  A  capitalist  sent  his  broker  $15400  to  lay  out  in  stocks, 

after  retaining  i  per  cent,   of  the  amount   purchased.' 
How  much  stock  did  he  purchase?        Ans.  $15801.00. 

18.  A  gentleman  laid  out  83025  in  stocks  which  were  10 
per  cent,  below  par.  What  was  the  nominal  value  of 
the  stock  purchased  ? 

19.  If  I  buy  coffee  at  30ct.  per  lb.,  and  sell  it  for  3Gct.  per 

lb.,  what  per  cent,  do  I  gain?  Ans.  20  per  cent. 

20.  A  merchant  bought  125  bushels  of  wheat  at  ^1.60   per 

bu.,  and  sold  it  at  a  profit  of  20  per  cent. ;    what  did 
hegetfi.rit?  Ans.  $210. 

21.  If  I  pay  $12000  for  a  house  and  lot  and  sell  them  at 
an  advance  of  25  per  cent.,  what  do  I  gain  by  the 
transaction  '^. 

22.  A  mei'chant  gave  $3.51  for  an  article  which  he  is  wil- 

ling to  sell  at  a  profit  of  88^  per  cent.,  how   must  he 
mark  it? 

23.  By  selling  a  tract  of  land  for  $4704  I  gain  12  per  cent, 
on  ic ;  how  much  did  it  cost  me  ?  Ans.  $4:i00. 

24.  If  ocwt.  ot  sugar  cost  $23.40,  what  will  lOcwt.  8qr.  cost? 

25.  A  merehantj  failing,  pays  only  60ct.  on  the  dollar  of  his 

indebtedness ;  how  much  will  a  man  receive  to   whom 
he  owes  $1800  ?  Ans.  $1080. 

26.  What  cost  4G2yd.  of  cloth  at  $1.06^  per  yd.? 

Ans.  $490.87i. 

27.  What  cost  83bu.  3pk.  2qt.  of  clover  seed  at  $8  per  bu.? 

205 


§261  AVERAGE. 


28.  What  per  cent,  of  $50  is  $G?  Ans.  12  pei-  cent. 

29.  What  is  115  per  cent,  of  $287.50  ?  Ans.  $330,625. 

30.  At  5  per  cent,  commission,  wkat   would  I  receive  for 
selling  $240  worth  of  property  ? 

31.  A  commission  merchant  sells  property  amounting  to 

$550.  Retaining  his  commission  of  5  per  cent.,  he 
lays  out  the  balance  after  deducting  a  commission  of 
2^-  per  cent,  on  the  amount  purchased.  How  much 
did  he  lay  out  ? 

32.  What  amount  can  I  retain  for  commission  at  8  per  cent, 
on  the  amount  invested,  if  I  have  received  $2647.08  ? 


AVERAGE. 


§  261.  The  average  of  two  numbers  is  one  half  of  their 
sum.     Thus,  the  average  of  7  and  13  is  (7  +  13)-^2=10. 

The  average  of  three  numbers  is  one  third  of  their  sum. 

The  average  o^  four  numbers  is  one  fourth  of  their  sum. 
And  so  on. 

§  262.  The  average  of  two  dates  is  a  date  lying  half  way 
between  them.  Thus,  in  any  year  June  23  is  the  average 
between  June  1  and  July  15. 

Ex.  1.  Find  the  average  of  2,  4.5,  5.75,  7,  and  9.25. 

2. 

45 

5.75  §  263.  Model. — Find  the  sum   of  the  five 

7.  given  numbers.  (§  159.)     Divide  this  sum  by 

9.25  5.  (§164.)     The  quotient  5.7  is  their  average. 

'    5)28.50 


5.7 
This  needs  no  explanation. 


206 


ALLIGATION    MEDIAL. 


^:^65 


Ex.  2.  What  if?  the  average  of  2,  3,  5,  and  6? 


:}.  What 

4.  What 

5.  What 

6.  What 

7.  What 
S.  What 
9.  What 

10.  What 

11.  What 

12.  What 

13.  What 

14.  What 

15.  What 


s  the  average  of  2,  5,  7,  and  10  1 

s  the  average  of  25  and  32  ? 

s  the  average  of  34  and  19  ? 

s  the  average  of  25,  32,  and  41  ? 

s  the  average  of  17,  29,  and  Go  ? 

s  the  average  of  25,  170,  and  195  ? 

s  the  average  of  2,  102,  111,  and  115? 

s  the  average  of  0,  5,  7.5,  25,  and  40  ? 

s  the  average  of  1,  7,  15,  25.25,  and  37.5  ? 

s  the  average  of  3,  7.5,  5.75,  11.75,  and  .625? 

s  the  average  of  20, 47,  35,  91.5,  79.5,  and  10.01? 

s  the  average  of  13, 15, 17, 29.5, 37.5,  and  63.75? 

s  the  average  of  0, 1,  7,  9,  25,  37,  and  39  ? 


ALLIGATION  MEDIAL. 


§  204.  This  name  is  given  to  the  process  of  finding  the 
moan  value  of  a  mixture,  when  the  values  of  the  substances 
composing  it  are  known. 

Ex.16.  If  41b.  of  sugar  worth  lOct.  per  lb.  are  mixed 
with  101b.  worth  12ct.  per  lb.,  what  is  a  pound  of  the  mix- 
ture worth?  • 

§  265.  Model. — Multiply 
lOct.  by4.  (§183.)  Multi- 
ply 12ct.  by  10.  (§  183.)— 
Add  the  products  together. 
(§179.)  Divide  the  sum  by 
14.  (§  185.)  The  quotient, 
llf  ct.,  is  the  average  price 
per  lb. 


4xl0ct.=   40ct. 
10x12"  =120" 

14  i60ct. 

14 

"20 
14 

6 


14 


llfct. 


207 


§266  AViCRAaE. 


ExPL\?fATio?^. — The  whole  mlstare  weighs  141b.,  whick 
evidently  cost  160ct.  :  and  1  fourteenth  of  this  amount  is 
the  average  price  per  lb. 

Rule. — Divide  the  whole  cost  by  the  number  of  articles  ; 
the  quotient  will  be  the  average  cost  per  unit. 

This  rule  applies  to  several  things  not  embraced  in  the 
definition. 

Ex.  17.  During  24  hours  the  thermometer  stood  for  2hi. 
at  55°,  for  3hr.  at  60°,  for  4hr.  at  65^  for  5hr.  at  70°,  for 
6hr,  at  75°,  and  for  4hr.  at  80^.  What  was  the  mean  tena- 
perature  of  the  day  ? 

18.  A  goldsmith  mixes  lOoz.  of  goid  16  cirats  fine  with  6oa. 

17  ci]*ats  fine  and  8az.  19  carats  fine ;  what  is^the  fina- 
nes^  of  the  mixture  ? 

19.  A  grocer  mixed  4gal.  of  wine  worth  $1  i  gallon,  5gal.' 
worth  $1.25  a  gallon.  ani'lOgal.  worth  $1.50  a  gallon  ; 
wh  (t  s^as  the  mixture  worth  per  gal.? 

20.  If  30gal.  of  molasses  at  lOct.,  40gii.  at  oOct.,  70gal,  at 
60ct.,  and  SOgal.  at  SOjt.,  be  mixed  to^^erjier,  wnat  ia 
a  gallon  of  the  mixture  worth  '/  Ans.  32Y\ct. 

21.  A  farmer  has  10  sheep  worth  $4  eaoh,  12  worth  $5 
eacli,  and  8  worth  $10  each  ;  what  is  their  averag« 
value  ? 


ALLIGATION  ALTERNATE. 


§286.  This  consists  in  finding  the  proportional  quantities 
of  several  simple  substances  which  shall  make  a  compound 
of  a  given  mean  value.  It  is,  therefore,'the  converse  of  the 
preceding.  • 

208 


ALT.IOATION    ALTFRNATE. 


§26T 


Ex.  22.  In  wbnt,  proportions  inir«t  sugars  worth  lOct., 
llct.,  13ct.,  and  ISot.,  be  mixed,  that  the  compuund  uiaj 
be  worth  14ot.? 

riO—  1         §  2C7.  Model.— Connect  10 


1     with  15,    11   wi'h   15,    and   13 
1    .with  15.      10  from  14  h-avf8  4; 


^^1  ^3.^  , 

L15ir_  .4  +  3  +  1  =  8     set  4  opposite  15:    11  from  14 

leaves  8;  ^et  3  opposie  15  :   13 

from  14  leaves  1  ;  set  1  opposite  15: — 14  tr<*iij  15  Uaves  1; 

pet  1  opposite  10,  11,  an«l  13.      ll«  nee  there  muft  he  lib.  ai 

lOct.,  lib.  at  Hot.,  and  lib.  at  13ot.,  to  81b.  at  15cf. 

Explanation. — After  arranging  the  several  prices  as  in 
the  model,  atid  placitig  the  mean  price  on  the  left,  we  con- 
nect each  price  below  the  mean  with  ont>  above  it,  and  t^ach 
price  above  the  mean  with  one  below  it.  Then  taking  the 
dififorence  between  each  price  and  the  mean,  we  set  this  dif- 
ference opposite  the  price  with  which  this  price  is  connect- 
ed ;  observing  during  the  operation  to  consider  all  the  prices 
as  abstract  numbers.  The  reason  for  all  this  is  evident 
when  we  consider  that  tach  pound  at  lOct.  falls  4ct.  below 
the  mean,  while  each  pound  at  15ct.  is  only  let.  above  it. 
To  average  these  two  values,  therefore,  we  mu.«t  have  41b, 
of  the  sugar  at  Inct,  to  every  one  at  l^cf.  For  a  siniilar 
reason,  it  requires  31b.  at  15ct.  to  counterbalance  lib.  afc 
llct.  And  as  the  mean  price  is  equidistant  between  13ct. 
and  15et.,  these  two  qualities  must  be  taken  in  equal  quan- 
tities. So  that  to  bring  the  three  i  ferior  qualities  up  to 
the  required  average,  it  is  neces^ary  to  take  4-|-3-f  1,  /.  <?., 
8lb.  of  the  superior  quality  to  lib.  of  each  of  the  inferior 
qualities. 

Rule.  I. — JJrrange  the  several  prices  in  a  vertical  coi- 
umriy  and  place  the  mean  price  on  the  left, 

N  209 


§267  AVERAGE. 


Connect  each  price  below  the  mean  with  one  above  it,  and 
each  price  above  the  mean  with  one  below  it. 

Find  the  difference  between  each  price  and  the  mean,  and 
set  it  opposite  the  price  luith  which  it  is  connected.  If  only 
one  difference  stands  opposite  any  price,  it  denotes  the  j)^o- 
portion  of  that  value  ;  but  if  several  differences  stand  oppo- 
site any  price,  their  sum  denotes  the  proportion  of  that  value. 

II.  If  it  is  required  to  have  a  specified  quantity 
OF  ANT  VALUE. — Find  the  proportions  as  above.  Then  say, 
As  the  proportion  found  for  this  value  :  the  quantity  reqiiir- 
ed  for  it  : :  the  proportion  for  any  other  value  :  the  quanti- 
ty required  for  it. 

III.  If  the  whole  quantity  oi'  the  mixture  is  speci- 
fied.— Find  the  proportions  as  ab  JVC.  Then  take  the  sum 
of  the  proportional  numbers,  and  say,  As  the  sum  of  the 
proportional  numbers  :  the  required  quantity  of  the   mixture 

::  the  proportion  for  any  value  '.  the  quantity  required  for 
that  value. 

Proof. — By  Alligation  Medial. 

Ex.  23.  In  what  proportions  may  gold  of  10,  13,  14,  and 
22  carats  fine,  be  mixed  so  that  the  compound  may  be  17 
carats  fine  ? 

24.  A  grocer  having  brandy  worth  $1  a  gallon,  wishes  to 
mix  it  with  water  so  that  he  can  sell  the  mixture  at 
80ct.  a  gallon.    In  what  proportions  must  he  mix  them  ? 

25.  In  what  proportions  may  liquors  worth  respectively  $1, 
$1.20,  $1.40,  and  $1.50  be  mixed,  that  the  mixture 
may  be  worth  $1.25  ? 

26.  A  farmer  wishes  to  mix  14bu.  of  wheat  worth  $1  per  bu. 
with  such  a  quantity  worth  $1.24  as  will  make  the  mix- 
ture worth  $1.03  ;  how  much  must  he  take  ? 

210 


EQUATION   OF   PAYMENTS.  §269 


27.  How  much  tea  at  80ct.,  70ct,j  and  60c  t.,  respectively, 
should  be  mixed  with  901b.  at  90ct.,  so  that  the  mix- 
ture may  be  worth  75ct.  per  lb.? 

28.  A  merchant  having  1001b.  of  sugar  worth  lOct.  per  lb., 
mixed  it  with  other  sugar  worth  respectively  5,  8,  and 
9ct.,  and  sold  the  mixture  at  S^ct.  How  much  of  each 
quality  was  there  in  the  mixture  ? 

29.  How  much  sugar  at  lOct.,  and  how  much  at  15ct.  per 
lb.,  must  be  taken  to  make  601b.  worjth  $7.20  ? 

Ans.  361b.  at  lOct.,  and  241b.  at  15ct. 
W,  A  grocer  mixes  1441b.  of  sugars  worth  respectively  12, 
10,  0,  and  4ct.  per  lb.,  and  sells  the  mixture  at  8ot. 
per  lb.  ,•  how  much  of  each  quality  does  he  take  ? 
31.  A  man  paid  S165  to  55  persons — men,  women,  and  boys; 
to  each  man  he  paid  $5,  to  each  woman  ^1,  to  each 
boy  50ct. ;  how  many  were  there  of  each  ? 

Ans.  30  men,  5  women,  20  boys. 


EQUATION  OF  PAYMENTS. 


§  268.  This  consists  in  finding  the  average  date  at  which 
several  amounts  due  at  different  times  may  all  be  paid,  so 
that  no  interest  shall  be  either  gained  or  lost. 

Ex.  32.  A  owes  B  S25  due  in  4mo.,  $50  due  in  6mo.,  and 
$75  due  in  8mo. ;  what  is  the  mean  time  of  payment? 

25x4mo.  =  100mo.  §269.  Model.— Multiply  4mo. 

50x6  "  =300  '^  by  25.    (§183.)     Multiply  6mo. 

75x8  "  =600  '-  by  50.     Multiply  8mo.  by  75.— 

150     15,0)100,0mo.  ^^^  ^^^  products  together.    Add 

— Wf- —  the  multipliers  together.    Divide 

Ogmo.  iQQQ^^Q^  Ijy  i^Q      rp^g  quotient 

6|mo.  is  the  mean  time  of  payment. 

211 


§269  AVERAGJ5. 

Explanation. — The  irifere^t  of  625  dollars  for  4  months 
IS  e(jn-il  to  the  interest  of  1  dollar  for  100  months:  the  in- 
fere.-st  of  350  for  6iiio.  =  the  interest  of  SI  forSOOuio.:  the  in- 
ter es  of  $75  tor  Sin!t.=:  the  interest  of  ^  I  for  GJOoOo  Hence 
the  interest,  of  the  several  am  >airth  for  their  respectiv;e  times 
Ln  equal  to  the  interest  of  $1  for  l0(»0ii!o.,  and  this  i»  equal 
to  the  interest  of  $150  for  Cfmo.  Oence  it  is  fair  that  tb« 
whole  amount  .should  be  paid  in  6|nio. 

IvUi.K. — Mnlfiplf^  e<i(}i  teiDi   of  cn'dlt  hij   the   number  oj 
Wtita  ill  ih>  correypititdiuij  juiijmeht,  iinil  divide  the   .sum  oj 
thf  pmdnrts  hy  the  aiim  of  (kr  niu(i'plitr»  :   the  quotient  wilf 
%e  ihe  medv  tim"  of  pai/mcnr,  ^ 

WjS^^  '"I-^.  a  Qian  owes  an  other   $500   due   in  8 mo.,  6400 
.dn»^  ii»  6mo.,  and  SGOO  due  in  9ino.  ;    what  is   the   averagj* 
term  of  er<t;dit  for  the  three  debt?*? 
84.   liou^ht  goods  a ■<  follows  :  ^h)0   on   a  credit   of  6rao.; 

,$200  on  o|tiio.;  and  $560  on  6mo. ;   what  average  credit 

f^hould  lie  allowed  me  on  the  «hole? 
B5.  B'U|ihi  $iOOO  worth  of  goods  lo  be  paid  for  as  follows; 

$200  on  the  day  of  purchase,  $400  in  5mo.,  and  $400 

in  I5m(>.      What  average  credit  &hould  be  allowed  in* 

ot)  the  whole  ? 

86.  Ill  what  time  should  the  f(*llowing  amounts  be  paid  ali 

at. once  :  $1000  due  in  5mo.,  $1200  in  Gmo.,  and  $1200 
in  8mo.? 

87.  I  owe  $100  to  be  paid  Jan.  15   $200  due  Feb.  15,  and 

$300  due  Mar.  9  ;  on  what  day  may  the  whole  debt  be 

paid  at  once  ? 

jV'o/*?  —  Felec'  }•  owe  day  from  wh'ch  the  periods  of  credit  nifty 
be  supi  08^•d  to  •  omnieiicts  In  this  i.saiKe.  Jan  15  in  the  mosi 
conv.uJtiit  Kind  ihe  interval  ehi|s:iiK  t>.  tweeti  tUis  date  wnd  each 
0*"  th**  o»h»  r«.  }0(d  then  proceed  according  to  the  rule.  Consider 
<a*ch  manth  80  days. 

212 


FQUATrOV    np  PATMFNTJ3.  §270 

ZH.  A  Tnai)  Awe^  liis  jieiii'hho'*  $12f)0  duo  in  8tn<>.  :  but  at 
the  <vi<l  of  3m').  be  pfi^ys  $250,  and  \\\  'Iwak  more  be 
pa)'  $'5'';  whut  oxteiisioti  of  credit  sboiild  be  allowed 
on  the  remainder  ? 

250x5nio.  =  l  50aM).  270.  M(.nKL.  — Multiply  5 

150x3  '*  =  450  "       mo.  b>  2:)!).  (^  1>^3  )    xMulti'pIj 

850)17007n7r.  ^'>»''.  *'>  ^^'^'     Add  tbe  pind- 

ucrs  t<tfri;rhor.    l)ivide  ITOOnio, 

'^"**'-  by  850.     The  quotient  2mo.  i« 
the  extension  of  cre<^it:. 

EXPL>NAT(()N.— The  debtor,  having  pqid  $2^0  (S  — 3) 
5mo  before  it  was  due,  is  entitled  to  a  credit  of  l25!)ino. 
on  $1  :  and,  having  paid  SI  50,  8'no.  before  it  was  due,  ii 
therefore  entitb'd  to  a  credit  of  45'hiio.  on  8'.  For  both 
prepayments  be  is  entirliMl  to  a  credit  eipiivMlent  to  $1  for 
ITOOmo.  Twe  remainder  unpaid  is  81250  — ($25!)  +  3l  r>0) 
=:$'*'50  :  nnd  a  credit  of  ITOOuio.  on  $1  is  equal  t<j  a  credit 
of  2mo    ofi  S85'>. 

Ex.  811.  I  owe  $1000  due  in  12mo.  Tf  T  pny  $100  at  the 
end  of  Bnin.,  ar.d  $100  at  the  end  of  4nio..  bow  long  be- 
yond the  r2ino.  should  my  creditor  wait  for  tb«  jiayment 
of  the 'balance  ? 

40.  I  owe  $2  100  due  in  6.no.     If  I  pny  8500  down,  ?3"0  i^ 
.2mo.,  and  S200  in  omo.,  in  bow  many  mOnth-i  from  thfc 

contraction  of  the  debt  should  I  pay  the  bjilnnce? 

41.  A  niercbaiit  owes  SI  200,  of  which  §200  is  to  be  paid  in 
4  mon'hs,  8400  in  10  mo-nths,  and  the  remainder  in  Id 
months:  if  he  pays  the  whole  at  ouce^  at  what  time 
rDu.-'t  he  ni'tke  the  payment? 

42.  A  merchant  owes  $ISOh  to  be  paid  in  1.2  months,  8240O 
to  be  [>aid  in  6  mou'hs,  and  82700  to.  be  p.iid  iu  0^- 
mouths:    what  is  the  averaj^e  tuue  of;  payment  ? 

2ia 


PROMISCUOUS   PROBLEMS. 


PROMISCUOUS   PROBLEMS. 


1.  Reduce  ,£19,  8|s.  to  pence. 

2.  Reduce  9oz.  16|-dwt.  to  grains. 

3.  Reduce  -  of  ~  of  IG-J-  to  its  simplest  form. 

■]r  12  ^  ^  . 

Ql.     Qi.  4i 

4.  Reduce  4-^,  -zrrr-,  tt-^  j  and  I  of -:^  to  their  least  com- 

'^     20f    26  -"        9 

mon  denominator. 

5.  Add  900.01,  450.037,  and  696.9  together. 

6.  Add  2y\,  6|-,  and  12if  together. 

7.  Add  -^  of  -*-  of  20,  -f-  of  -f-  of  24^,  and  ^,-  of  2-i-  together. 

6a  2i 

8.  From  -^  take  -^. 

9.  From  -f-  of  f  of  3i  take  f  of  f. 

10.  From  $49f  take  i4.75  +  $5^4-$9.30. 

8-1-         17 

11.  Multiply  -4  by  -7-. 

5 

17 

12.  What  is  the  product  of  -fi  by   ---  ? 

50     J    2Q_:2. 

5 

13.  Multiply  ^g-^^  by  -^V  of  if  of  5f 

14.  Divide  f  of  3-|  by  -f-  of  6i. 

15.  Divide  1301|-  by  161.3. 

16.  Divide  $1843i  by  368f, 

17.  What  is  the  insurance  on  $3125  at  6i  per  cent.? 

18.  A  commission  merchant  sold  19  firkins  at  45ct.  per  lb., 

and  retained  5  per  cent.,  commission  ;    how  much   did 
he  return  to  the  owner  ?  ^ 

.19.  What  is  the  par  value  of  two  certificates  of  stock;   one   ^ 

214 


PROMISCUOUS   PROBLEMS. 


for  S350  at  2}  per  cent,  discount,  the  other  for  $527-.- 
60  at  5  per  cent,  advance  ? 

20.  What  is  the  amount  of  ^1054,  10s.  9d.  for  2yr.  9mo.  at 
4  per  cent,  per  annum  ? 

21.  If  the  interest  of  a  certain  amount  of  money  at  6  per 
cent,  is  $241.80,  what  is  the  interest  of  the  same  sum 

,      for  the  same  time  at  7^-  per  cent.? 

22.  At  what  rate  per  cent,   per  annum   will  jei8295  10s. 
amount  to  £1898,  2s.  l^d.  in  9  months] 

23.  What  is  the  greatest  common  measure  of  560,  880, 
1028,  and  1296? 

24.  What  is  the  least  common  multiple  of  36,  18,  33,    11, 
and  6  ? 

25.  What  is  the  greatest  common  measure  of  56,  154,  and 

182? 

26.  What  is  the  least  common  multiple  of  2,  4,  10,  7,  14, 

15,  and  21? 

27.  Resolve  528  into  its  prime  factors. 

28.  What  prime  factors  are  common  to  360,  420,  and  840  ? 

29.  I  sold  125A.  2R.  20P.  of  land  for  ^2050  ;  how  much 
did  I  gain  or  lose,  if  I  gave  $15.50  per  A.  for  the  land? 

30.  I  bought  a  lot  of  English  paper  for  .£698,  10s.  6d.,and 
sold  it  at  a  proJGit  of  75  per  cent. ;  how  much  did  I  re- 
ceive for  it  in  Federal  currency  ? 

31.  What  is  the  amount  of  £300,  10s.  for  2yr.  3mo.  at  in- 
terest compounded  semi-annually,  at  8  per  cent,  per 
annum  ? 

32.  What  is  the  square  root  of  509796? 

33.  What  is  the  cube  root  of  16003008  ? 

34.  Find  the  greatest  common  measure  of  1538,  2307,  and 

3845. 

215 


PROMISCUOUS   PROBLEMS. 


35.  I  sold  i  of  my  land  to  A,  ^  of  ifc  to  B,  and  retained 
20i.'A.  for  myself;  how  much  had  I  at  first? 

ii6.  A,  B,  and  C  trade  in  p'^rtnersliip.  A  invests  SIOOO 
'for  12  mon'hs;  B,  fj^l 500  for  10  months  ;  and  C,  ?2000 
for  0  months.  How  bhali  their  profit  of  SlUOO  be  di- 
vided ? 

37.  How  many  barrels  of  potatoes  at  S2.50  per  bbl.  should 
be  exchanged  for  a  Intgshead  of  sngnr  weighing  IST-^^lb. 
groBP,  worth  115.00  a  hundred  pounds  net,  tare  being 
8  per  cent.? 

8?.  How  many  firkins  of  butter,  at  25ct.  per  lb.,  can  bo 
bought  fur  9iiiO.  interest  of  §800  at  7  per  cent,  per 
annum? 

89.  Having  been  engaged  in  merchandise  with  a  capital  of 
$19500,  I  realized  a  profit  of  33^^  per  cen^t.,  which  I 
immediately  inve>ted  in  land  at  $16.50  per  A.;  how 
many  acres  did  I  buy  1 

40.  If  [  owe  three  notes,  one  for  ^630  due  3mo.  hence,  an 
other  for  ^800  due  6mo.  h'nce,  and  the  o-her  for  Si  000 
due  15rao.  hence,  in  what  timjc  might  I  fairly  pay  the 
three  notes  together? 

41.  If  i2lb.  of  tea  @81.->0,  151b.@$1.44,  and  18ib.@S1.80, 
be  mixed  together,  what  irs  liae  value  of  111b.  of  the 
ntixture  ? 

42.  What  i.s  the  4th  power  of  7^? 

43.  What  is  the  cube  of  3.5? 

44.  What  is  the  c.ibo  root  of  ]9.Gi»3? 
45    What  is  the  8(iuare  root  of  7l.)3^g  ? 

46*.  Required  to  fill  a  hogshead  with  two  kinds  of  wine 
worth  $1.20  and  $1.05  fier  gal.  respectively,  so  that  the 
mixture  wil'  be  worth  ?I.15  per  gal.;  how  mariy  gal- 
lons of  each  kind  will  be  rcijuired  ? 

216 


pnOMISrrOUS   P'CtBLFMS. 

47.  The  total  Mock  in  n  UnilnKaJ  is  SIOOOoOO  tb'^  net  m- 
cctme  for  a  veur  is  $501'00 ;  wlia  dividend  will  I  re- 
ceive f  r  SlOOOn  woith  of  .-tuck  ? 

18    I  exebaficed  a  houf^e  jmd  lot  worth  £500,  15^  fnr  land 

*  «  •       ? 

worth  SIU.50  per  A.;   how  tuuch    l;iud   did    I   r^'C'•ue  • 

49.  A  pedlar  exclxintred  a    pit-ce  of  calico,   rated   at    2 let. 

per  yd.,  ff^r  a  fii  kin    of  butter   worih    '2'2}ji5t.    per    lb; 
how  many  yards  of  calico  were  tb  re? 

50.  I  imported  V):")?.  of  iron  wor-b  S82  per  T, ;  what  was 
th«  duty  o!i  it  at  3*3^  |>er  c<Mit.? 

51.  ^Vlnjt  is  il»e  net  weight  ot*  275  bags  of  coffee,  weighing 
each  78tb.  irross,  tnre  being  4  per  cent.? 

.Vi.  What  cost  7UA.  nn.  25P.  of  land  at  $.\5.75  per  A.? 
33.   What  co.-t  51'.  IGcwf.  3.|r.  o    iron  at   $1:.125   per   cwt.? 
i4.  If,  by  se'ling  a  tract  of  land    for   St)450,    I   lo>=e   4    per 

cent,  of  what  it  cost  nie,  for  what  would  I  have  had  to 

sell  it,  to  jjuin  C^\  f>^»r  cent.? 
55.   Bought  2')T.  IGcwr.  of  iron  a^  =£1  1,  1  Gs.  per  T.  ;  and    oM 

the  whole  for  32 JOO  ,   what  did  [  gaiu  or    lose   p«r    T? 
^iiy^   What  IS  the  present  worth  of  $-00n,  due  in    2yr.    Huio. 

1.')  Ia.»  ijitere:-t  beiiig  at  fi  per  cetit.  p^r  annum  r* 

57.  What  is  the  discount  on  £iXO^  due  in  Syr.  Omo.,  inter- 

est hfiiig  at  8  per  cent,  per  annuu)  ? 

58.  I  wish  to  borrow  81  150  iti  bank  :    interest   being   at    6 

percent,  per  tinnum,    what    ujust    be    the   fa<^e   of  t!) ; 
proper  note  at  90  days? 

59.  In  what  tiute  will  £4o2,  15s.,  at  C  percent,  per  annum, 

amount  to  £502,  1 1  -.  '^d.? 
6D.  If  the  insurance  of  :B25ajO  is  S400,  wb-it  is  the  rate  per 

C'-'Iit.? 

($1.  What  per  cent,  of  00  is  1,25? 
Q2.  Wluit  per  cent,  of  75  is  125  ? 

217 


PROMISCUOUS    PROBLSMS, 


63.  What  is  S-^\r  per  cent,  of  S11755  ? 

64.  If  5|-A.  of  land  cost  $144.50,  what  will  17A.  3E.  19.- 
375P.  cost  ? 

65.  If  1.37gal.  of  sorghum  molasses  cost  $1.4375,  what  will 
I3.7gal.  cost? 

66.  Divide  17mi.  5fur.  25rd.  by  1.5. 

67.  Multiply  3deg.  17min.  45sec.  by  2.03. 

68.  Dividend  is  £1,  18s.  9.5d.,  divisor  is  4.9,  what  is  the 
quotient? 

69.  Dividend  is  Ibu.  3pk.  4.5qt.,  e^uotient  is  75bu.  2pk, 
4qt.,  what  is  the  divisor  ? 

70.  From  1.475T.  take  17cwt.  Iqr.  19.29lb. 

71.  Add  together  4.75gal.,  3.07qt.,  7.45pt.,  and  6.19g). 

72.  What  cost  17bbl.  flour  at  $10  per  1001b.,   3bu.  salt  at 

$1.25  per  bu.,  and  677.51b.  pork  at  $0,065  per  lb.? 

73.  If  75  persons  eat  800bu.  corn  in  1  year,  how  long  will 

600bu.  last  90  persons  ? 

74.  If  150  copies  of  a  book  of  200  pages  require  6rm.  4qr, 
of  paper,  how  many  reams  will  15000  copies  of  a  book 
of  224  pages  require  ? 

75.  If  Irra.  of  paper  weigh  301b.  and  cost  30ct.  per  lb., 
what  will  the  paper  cost  for  an  edition  of  1000  copies 
of  a  book  which  requires  5rm.  lOqr.  for  96  copies? 

76.  If  83iT.  of  coal  cost  $405.50,  what  will  17T.  3cwt.  Iqr. 
bring  at  16|-  per  cent,  advance  ? 

77.  The  second,  third,  and  fourth  terms  of  a  proportion  are 
-|,  1^,  and  2.5,  respectively ;  what  is  the  first  term  ? 

78.  If  the  first,  third,  and  fourth  terms  of  a  proportion  are 
$64.96,  7cwt.  Iqr.,  and  4cwt.  2qr.,  respectively,  what 
is  the  second  term  ? 

79.  Multiplicand  is  94;  product  is  .66;  what  is  the  multi- 
plier ? 

218 


PROMISCUOUS   PROBLEMS. 


80.  I  bought  6251b.  of  cheese  for   $62.50,  and  sold  it  at 

12ict.  per  lb.  ;  how  much  per  cent,  did  I  gain? 

81.  I  own  I  of  a  ship  worth  820000,  and  have  insured  it  at 
2.375  per  cent. ;  what  insurance  do  I  pay  ? 

82.  What  is  the  amount  of  $2169.845  for  lyr.  lOmo.  17da. 
at  7  per  cent,  per  annum  ? 

83.  What  are  the  prime  factors  of  7825? 

84.  What  are  the  common  prime  factors  of  875  and  1750  .'' 

85.  How  many  hours  will  there  be  in  the  year  11)00  ? 

86.  The   Mecklenburg  Declaration    of   Independence    was 

made  May  20,  1775;  North  Carolina  unanimously 
seceded  from  the  United  States  May  20,  1861 ;  how 
many  days  elapsed  between  these  two  great  events  1 

87.  What  cost  30001b.  of  corn  at  S3.00  per  bbl.] 

88.  What  cost  5.25bbl.  of  flour  at  $  .04  per  lb.? 

89.  An  officer,  in  pursuit  of  a  criminal,  goes  lOmi.  per  hr.; 
the  criminal,  who  has  36mi.  the  start,  goes  7mi.  per  hr.; 
how  far  must  the  officer  go,  to  catch  the  criminal  ? 

90.  Bought  40gal.  wine  at  $2.50  per  gal. ;  lost  5gal.  by 
leakage  :  how  must  I  sell  the  remainder  per  gal.  so  as 
to  gain  25  per  cent,  on  the  whole  ? 

91.  A  vessel  laden  with  3000bu.  wheat,  found  it  necessary 
to  throw  25  per  cent,  of  her  cargo  overboard ;  what 
was  her  loss  at  $1.25  per  bushel  1 

92.  What  is  the  value  in  Avoirdupois  weight  of  161b.  5oz. 

lOdwt.  12gr.  Troy? 

93.  How  many  sheets  in  7  reams  of  paper? 

94.  If  7  silver  spoons  weigh  lib.  2oz.  3dwt.,  what  will  each 
spoon  weigh  ? 

95.  If  2A.  produce  45bu.  3pk.  6c][t.  Ipt,  of  corn,  how  much 

will  32A.  produce  ? 

219 


PR.OM.'SCUOUS  TRrELEMS. 


96.  A.li  together  i  of  i  of  an  Acrp,  75^^?.  -fll.,  and  |A.r 

97.  What  part  f.f  u  futhom  is  Sifr.V 

98.  What  is  the  anionnt  of  $30U0  for  Gmo.  24Ja.  at  7^-  per 
ceiif.  per  arirnim  ? 

99.  A  ari.l  H  [)iirclj;iPe(l  a  house  for  $3000,  of  which  A  paid 
31800,  how  shall  they  divide  a  rent  of  $350  ? 

iOO.    Wliat  it^  the  scjuare  root  of  57(5? 
101.   What  is  the  4rh  root  of  GoGl  ? 
I'v2.   WhMt  is  the  cube  root  of  yf-^iouo  ^ 

103.  How  much  stock  at  7  perceut.  advance  niaj  be  bought 
fur  Sf)350? 

104.  Ij.'uglit  lOrm.  of  paper  at  ?3.50  per  rm.,  and  sold  it 
at  $  .25  per  quire,  h  -vv  luuch  did  I  gain  or  lose  on  it  all? 

105.  Bought  30ubhl.  of  flour  for  $2250,  sold  I  of  it  at  $0 
per  hbl.,  jiiid  the  remainder  at  $8  per  bbh,  how  much 
did  T  receive  for  the  whole? 

106.  Reduce  26^  to  a  decimal  form. 

107.  Miihiply  four  rhousjindths  by  five  hundredth!*. 

108.  Multiply  fdur  hundred  ati({  fifty   by   two  hundredths. 

109.  Diviile  -even  tenths  by  one  hundredth. 

liO.  \\''hai  is  the  difference  between  thirty-five  hundredths^ 
and  thirty-five  thouHa'jdrh.s  ? 

111.  What  is  the  2nd  term  of  h  proportion  whcse  T.-t,  3rd^ 
atid  4(h  terms  are  7,  1.3,  and  19,  respectively  ? 

112.  If  one  jicre  of  lar»d  costs  jC2,  15s.  4d.,  what  will  be  th» 
cost  (.f  173 A.  2R.  14P.  at  the  same  rate  ? 

113.  A  geijtlemah's  estate  is  worth  dC42  !5,  Is.  a  year  :  what 
may  he  spend  per  day  and  yet  save  i^lOOO  per  annum? 

114.  A  father  left  his  son  a  foitune,  -^  of  which  he  ran 
through  in  8  months,  -^  of  the  remainder  lasted  him  12 
months  longer,  when  he  had  barely  £820  left:  what 
hum  did  his  father  leave  htm  ? 

220 


PROMlSCUOye   PROBLEMS. 


115.  Ther?  are  1000  men  be>ie«^ed  in  a  to'-vn  with  j)rovi" 
einiis  for  5  weeks,  allowing  each  uian  10  ounces  a.  day. 
If  rh-y  art>  reinf.rrced  by  500  more  and  no  relief  can 
be  afforded  til!  the  end  of  8  wi-ek.s,  Low  many  ouiKces 
n\\ii\t  be  given  daily  to  each  man  '{ 

116.  A  fjitlier  pave  -{^  of  his  eMate  to  one  Hon,  and  ^'^  of 
the  remainder  to  another,  leaving  the  rest  to  his  widow. 
The  difference  of  the  childjen'w  h-gaeies  was  £514,  6s. 
Si\.  :    what  was  the  widow's  portion  ? 

117.  If  Hcwt.  2(jr.  of  sugar  cost  S12l\92,  what  will  be  tlie 
price  of  Ocwt.? 

118.  If  the  freight  of  80  tierces  of  Kiigar,  each  weighing 
Sicwt..  150. miles,  cost  S8[,  what  must  be  paid  fur  the 
freight  of  30hhd  of  sugar,  each  weighing  12cwl.,  50 
miles  ? 

119.  If  one  pound  of  tea  be  equal  in  value  to  50  ornnges, 
nnd  70  oranges  be  worth  8*  lemons,  what  is  the  value 
of  a  pound  of  tea  when  a  lemon  is  worth  2  cents? 

120.  If  60  bushels  of  oats  wilbserve  2(  horses  f.r  40  dajR, 
how  long  will^iO  bushels  swerve  48  horses  at  the  same 
rate  ? 

131.  W!,at  will  be  the  cost  of  2hhd.  5ga].  3qt.  2gi.  of  mo- 
Jasse^,  at  12}  cents  p^r  (jnart? 

122.  Wliat  is  the  interest  of  53153.82  for  0  years,  at  4} 
per  cent,  per  annum  ? 

123.  What  is  the  interest  of  831573.25  for  10  months  at  6 
per  cent,  per  annum  ? 

124.  What  will  be  the  amount  of  $9537.15  for  11  years,  2 
months,  and  18  days  at  7  per  cent,  per  annum  ? 

125.  What  will  be  the  amount  of  83758.56  for  3  years  at 

7  per  cent.,  th©  interest  being  compounded  semi-annu- 
ally ? 

221 


PROMISCUOUS   PROBLEMS. 


126.  If  I  buy  895  gallons  of  molasses  and  lose  it  per  cent, 
by  leakage,  how  much  have  I  left  ? 

127.  Bought  a  piece  of  cloth  containing  150  yards  for  $650: 
what  must  it  be  sold  for  per  yard,  in  order  to  gain  $300? 

128.  What  is  the  bank  discount  on  a  note  of  $556.27  pay- 
able in  60  days,  discounted  at  6  per  cent,  per  annum  ? 

129.  The  sum  of  two  numbers  is  5330,  their  difference 
is  1999  :  what  are  the  two  numbers  ? 

130.  How  many  scholars  are  there  in  a  class,  to  which  if  11 
be  added  the  number  will  be  augmented  one-sixteenth? 

132.  Sound  travels  about  1142  feet  in  a  second.  If  then 
the  flash  of  a  cannon  be  seen  at  the  moment  it  is  fired, 
and  the  report  heard  45  seconds  after,  what  distance 
would  the  observer  be  from  the  gun  ?  • 

133.  What  number  is  that  which  being  augmented  by  85, 
and  this  sum  divided  by  9,  will  give  25  for  the  quotient? 

134.  One -fifth  of  an  army  was  killed  in  battle,  i  part  was 
taken  prisoners,  and  J^  died  by  sickness  :  if  4000  m.en 
were  left,  how  many  men^did  the  army  at  first  consist  of? 

135.  The  greatest  of  two  numbers  is  15.  and  the  sum  of 
their  squares  is  346 :    what  are  the  two  numbers  ? 

136.  At  what  rate  per  cent,  will  $1720.75  amount  to  $2325.- 
86  in  7  years  ? 

137.  In  what  time  will  $2377.50  amount  to  $2852.42  at  4 
per  cent,  per  annum  ? 

138.  What  principal  put  at  interest  for  7  years,  at  5  per 
cent,  per  annum,  will  amount^to  $2327.89? 

139.  What  is  the  greatest  common  measure  of  945,  1660, 
and  22683  ? 

140.  What  is  the  greatest  common  measure  of  204,  1190, 
1445,  and  2006  ? 

141.  Find  the  least  common  multiple  of  6,  9,  4, 14,  and  16. 

222 


PROMISCUOUS   PROBLEMS. 


142.  What  is  the  least  common  multiple  of  11,  17,  19,  23, 
and  7? 

143.  What  is  the  least  common  multiple  of  7,  15,   21,  28, 
35,100,125? 

144.  Reduce  ^-^^*-  to  a  mixed  number. 

145.  Reduce  149|^  to  an  improper  fraction. 

146.  Reduce  375|^  to  an  improper  fraction. 

147.  Reduce  174947^^^ f|-g  to  an  improper  fraction. 

148.  Reduce  fig-  to  its  lowest  terms. 

149.  Reduce  -tt-Vt  *^  ^^^  lowest  terms. 

150.  Reduce  14-1^-  to  its  lowest  terms. 

151.  Reduce  %,  -^,  and  -j\-  to  their  least  common   denomi- 
nator. 

152.  Pteduce  ^V?  ^\y  ^^^  v  *^  their  lea.<^t  common  denomi- 
nator. 

153.  Find  the  least  common  denominator  and  add  the  frac- 
tions, yV?  h  h  and  t, 

154.  Find  the  least  common  denominator  and  add  /^,  }f  ^-, 
and  -V- 

155.  Multiply  5^-  by  ■}. 

156.  Multiply  f5  by  |  of  9. 

157.  If  80  yards  of  cloth  cost  $340,  what  will   050  yards 
cost  ? 

158.  If  120  sheep  yield  330  pounds   of  wool,  how   many 
pounds  will  be  obtained  from  1200  sheep  1 

159.  If  6  gallons  of  molasses  cost  SI. 95,  what  will  6  hogs- 
heads cost  ? 

160.  If  *-  of  a  yard  of  cloth  cost  ^1|,  what  will  7^  yards 
cost  ? 

161.  What  is  the  cost  of  28|-  yards   of  cloth,  at  $4f  per 
yard  ?  i 

223 


PUOMISrUOUS    PROBLTfMS. 


162.   Whnt.  is  the  interest  of  $1914.16  for   18   years  at   3^ 

per  c#Tit.  per  animni  ?' 
ICS.   What  is  the  a»n>uni:,  of  $7958.70  for  9   raonths  at  (^ 

por  ctMit    per  unniuij  ? 

164.  A  Fnerch-mt  h-d^  1-00  barrels  of  fl  )ur  ;  he  shipped  64 
per  c^uf.  of  it  and  sold  the  remainder:  how  much  did 
lie  sell? 

165.  Two  men  had  eacli  $240.  Que  of  them  Rpend8l4per 
cent.,  and  the  otiier  1<Sl  per  cent.  :  how  ir)a[»y  dollar* 
more  ditJ  one  >pend  than  the  n<her? 

166.  Wh;)t  is  the  diife'vnce  between  f)^-  per  cent,  of  $800 
and  6*-  per  cent    of  Sl<'50  ? 

167.  What  is  the  square  root  of  1.519:^^9!? 

168.  What  is  the  Hj.nn-e  root,  of  :^'^:i7296l  ? 
1C>9.   What  is  the  cube  r  .ot  of  18  28541  ? 

170.  What  is  the  cnhe  root  of  2;<>5408-6008  ? 

171.  if  a  portion  re(;='ivt\s  ?1  iur  ±  of  a  djy's  work,  how 
innrh  is  that  a  d^y  i 

112.  \V\u\i  nmnher  \<i  rh»f  of  whicii  i,  -\,  and  |-  added  to- 
gether, will  muke  6-3? 


Date  Due 


tilG 


Tir«~ 


L.  B.  Cat.  No.  1 137 


\ 


511.02   L255  23448 


r^Ouri  Own  Series  of  School  Books.l 


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